AN INVESTIGATION RELATING LONGITUDINAL PILOT-INDUCED ... · 4-1. MAX GAP (LAMARS) Nichols Charts for Cases B, N, W and Y.....4-3 4-2. MAX GAP (LAMARS) Phase 2 Sum-of-sines Task Data.....4-6

DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY

AIR FORCE INSTITUTE OF TECHNOLOGY

Wright-Patterson Air Force Base, Ohio

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

AN INVESTIGATION RELATING LONGITUDINAL PILOT-INDUCED

OSCILLATION TENDENCY RATING TO DESCRIBING FUNCTION

PREDICTIONS FOR RATE-LIMITED ACTUATORS

THESIS

Joel B. Witte, Major, USAF

AFIT/GAE/ENY/04-M16

The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government

AFIT/GAE/ENY/04-M16




THESIS

Presented to the Faculty

Department of Aeronautics and Astronautics

Graduate School of Engineering and Management

Air Force Institute of Technology

Air University

Air Education and Training Command

In Partial Fulfillment of the Requirements for the

Degree of Master of Science in Aeronautical Engineering

Joel B. Witte, BS

Major, USAF

March 2004

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

AFIT/GAE/ENY/04-M16




Joel B. Witte, BS Major, USAF

Approved: __________________________________ ______________ Bradley S. Liebst (Chairman) date __________________________________ ______________ Richard G. Cobb (Member) date __________________________________ ______________ Russell G. Adelgren (Member) date

iv

AFIT/GAE/ENY/04-M16

Abstract

The purpose of this study was to investigate pilot-induced oscillations (PIO) and

determine a method by which a PIO tendency rating could be predicted. In particular,

longitudinal PIO in the presence of rate-limited actuators were singled out for

examination. Sinusoidal input/triangular output describing function techniques using

Nichols charts were used. A new criterion dubbed Gap Criterion was calculated for PIO

sensitivity. This criterion consists of the product of additional pilot gain and the

normalized maximum amplitude of the commanded actuator necessary to cause PIO.

These results were paired with simulator and flight test PIO tendency rating data. The

PIO rating scale used was the PIO tendency classification of MIL-HDBK-1797. This

concept was applied to two historical test databases, HAVE PREVENT and HAVE

OLOP. Additional PIO data was gathered in the Large Amplitude Multimode Aerospace

Simulator (LAMARS) at the Air Force Research Laboratory (AFRL), Wright-Patterson

AFB, Ohio and the USAF NF-16D Variable In-flight Stability Test Aircraft (VISTA) at

Edwards AFB, California. Correlation between PIO tendency rating and Gap Criterion

was determined for each dataset. Most datasets exceeded a confidence level of 95% that

a correlation existed. Follow-on analysis for better curve fitting was accomplished; a

logarithmic fit was judged best. Datasets were combined with success to demonstrate the

universality of the Gap Criterion for correlating PIO tendency ratings for longitudinal

PIO involving rate-limited actuators.

v

Acknowledgements

The Air Force Institute of Technology and the United States Air Force Test Pilot

School cosponsored this study. I would like to thank the following people for their

efforts and contributions: My thesis advisor, Dr Brad Liebst, who proposed this study and

offered key insights; Mr. Andy Markofski and Mr. Mike Steen of General Dynamics

Advanced Information Systems who programmed the aircraft dynamics for the VISTA

NF-16D; the VISTA maintenance team who worked miracles to return the aircraft to

flying status; Major Russ Adelgren and Mrs. Cynthia Roell, who made sure the flight test

portion succeeded and Mr. Curt Clark and Mr. Jeff Slutz who worked diligently to ready

the LAMARS simulator for the simulator test portion.

I’d especially like to thank the members of the MAX GAP test team who made

this project succeed: Captain Erik Monsen, Royal Norwegian Air Force; Captain Thomas

Washington, USAF; Captain Kris Cowart, USAF and Captain Rich Salasovich, USAF.

I’d also like to thank Major James Less, our Test Pilot School staff sponsor, and Major

Rick Palo who survived multiple PIO sorties as safety pilot in the VISTA F-16.

Most importantly, I’d like to thank my wife for her support, love and

understanding through the trials of the past two and a half years.

vi

Table of Contents

Page

Abstract .............................................................................................................................. iv

Acknowledgements............................................................................................................. v

Table of Contents............................................................................................................... vi

List of Figures .................................................................................................................... ix

List of Tables .................................................................................................................... xv

List of Notations and Symbols........................................................................................ xvii

I. Introduction ............................................................................................................... 1-1

General......................................................................................................................... 1-1 Background.................................................................................................................. 1-2

PIO Defined. ............................................................................................................ 1-2 PIO History. ............................................................................................................. 1-4 PIO Categories. ........................................................................................................ 1-6 PIO Category II........................................................................................................ 1-7 PIO Scales................................................................................................................ 1-8 PIO Prediction Methods......................................................................................... 1-10 Gap Criterion. ........................................................................................................ 1-11

Objectives .................................................................................................................. 1-11 Approach.................................................................................................................... 1-12 Scope.......................................................................................................................... 1-12

II. Theory ...................................................................................................................... 2-1

Describing Function Development .............................................................................. 2-1 Saturation Nonlinearity Describing Function.............................................................. 2-3 Closed Loop Describing Function Approximation...................................................... 2-4 Sinusoidal Input/Triangle Output Describing Function Approximation ..................... 2-7 Applying Describing Function Results to Predict PIO.............................................. 2-11 Pilot Model ................................................................................................................ 2-13 Gap Criterion............................................................................................................. 2-16

Gap Criterion Formulation..................................................................................... 2-17 Type I. .................................................................................................................... 2-19 Type II.................................................................................................................... 2-20 Type III. ................................................................................................................. 2-21 Type IV. ................................................................................................................. 2-21

vii

Page

Example of Gap Criterion Application ..................................................................... 2-21 III. Analysis of Selected Historical Data ...................................................................... 3-1

HAVE PREVENT Analysis ........................................................................................ 3-1 Gap Criterion Calculation for HAVE PREVENT Datasets..................................... 3-4 HAVE PREVENT Gap Criterion Summary............................................................ 3-7 HAVE PREVENT Gap Criterion Correlation. ........................................................ 3-8 HAVE PREVENT Best Curve Fit. ........................................................................ 3-11

HAVE OLOP Analysis.............................................................................................. 3-14 Gap Criterion Calculation for HAVE OLOP Datasets. ......................................... 3-15 HAVE OLOP Gap Criterion Summary. ................................................................ 3-19 HAVE OLOP Gap Criterion Correlation............................................................... 3-19 HAVE OLOP Best Curve Fit................................................................................. 3-21

HAVE PREVENT and HAVE OLOP Summary ...................................................... 3-24 IV Analysis of Project MAX GAP (LAMARS) Data ................................................... 4-1

MAX GAP (LAMARS) Analysis................................................................................ 4-1 Gap Criterion Calculation for MAX GAP (LAMARS) Datasets. ........................... 4-2 MAX GAP (LAMARS) Gap Criterion Summary. .................................................. 4-5 MAX GAP (LAMARS) Gap Criterion Correlation................................................. 4-5 MAX GAP (LAMARS) Best Curve Fit................................................................... 4-8

MAX GAP (LAMARS) Summary ............................................................................ 4-11 V. Analysis of Project MAX GAP (VISTA) Data........................................................ 5-1

MAX GAP (VISTA) Analysis..................................................................................... 5-1 Gap Criterion Calculation for MAX GAP (VISTA) Datasets. ................................ 5-2 MAX GAP (VISTA) Gap Criterion Summary. ....................................................... 5-6 MAX GAP (VISTA) Gap Criterion Correlation. .................................................... 5-6 MAX GAP (VISTA) Best Curve Fit. ...................................................................... 5-8

MAX GAP (VISTA) Summary ................................................................................. 5-10 VI. Analysis of Combined Gap Criterion Data ............................................................ 6-1

LAMARS Combined Datasets .................................................................................... 6-1 VISTA Combined Datasets ......................................................................................... 6-3 LAMARS and VISTA Combined Dataset Correlation ............................................... 6-4 Combined Dataset Analysis and Observations............................................................ 6-5

VII. Conclusions ........................................................................................................... 7-1

viii

Page

Appendix A. Matlab/SimulinkTM Code ......................................................................... A-1

Appendix B. State-Space Matrices for Project MAX GAP........................................... B-1

Appendix C. Correlation Computation .......................................................................... C-1

Appendix D. MAX GAP Histograms ............................................................................ D-1

Bibliography ............................................................................................................... BIB-1

Vita........................................................................................................................... VITA-1

ix

List of Figures

Figure Page 1-1. YF-22A Accident Sequence (Hodgkinson, 1999:128)............................................ 1-4 1-2. Actuator Saturation Example Using an Input = A sin(ωi t) ..................................... 1-8 1-3. SimulinkTM Actuator Model .................................................................................... 1-8 1-4. PIO Tendency Classification (MIL-HDBK-1797, 1997:152)................................. 1-9 2-1. Example of a Nonlinear System .............................................................................. 2-1 2-2. Saturation Nonlinearity and the Corresponding Input-Output Relationship (Slotine

and Li, 1995:173)..................................................................................................... 2-3 2-3. Actuator Model Development (Klyde and others, 1995:22) ................................... 2-5 2-4. Closed Loop Actuator Transfer Function Diagram................................................. 2-6 2-5. Rate-Limiting Input and Output (Klyde and others, 1995:42-46)........................... 2-8 2-6. Describing Function Phase Angle Comparison..................................................... 2-10 2-7. Pitch Tracking Closed Loop System ..................................................................... 2-11 2-8. Simplified Pitch Tracking Closed Loop System ................................................... 2-11 2-9. Neal-Smith Pilot Model Constraints...................................................................... 2-15 2-10. Pitch Tracking Closed Loop System for Gap Criterion...................................... 2-16 2-11. Four Resulting Gap Criterion Types................................................................... 2-18 2-12. Case I Effective Pilot Gain Increase.................................................................... 2-19 2-13. Case II Effective Pilot Gain Decrease ................................................................. 2-20 2-14. Nichols Chart of the Example Problem............................................................... 2-22 3-1. Phase 2 Sum-of-sines Pitch Tracking Task ............................................................. 3-2 3-2. Phase 3 Discrete HUD Pitch Tracking Task ........................................................... 3-2

x

Figure .............................................................................................................................Page 3-3. Phase 3 Target Tracking Task ................................................................................. 3-3 3-4. Longitudinal State Space Diagram (Matlab/SimulinkTM, 2001) ............................. 3-4 3-5. HAVE PREVENT Nichols Charts for Cases A, B and C ....................................... 3-6 3-6. HAVE PREVENT Phase 2 Sum-of-sines Task LAMARS Data ............................ 3-9 3-7. HAVE PREVENT Phase 3 Discrete HUD Pitch Tracking Task LAMARS Data .. 3-9 3-8. HAVE PREVENT Phase 3 Target Tracking Task LAMARS Data...................... 3-10 3-9. HAVE PREVENT Phase 2 Sum-of-sines Task LAMARS Data with Logarithmic

Curve Fit ................................................................................................................ 3-12 3-10. HAVE PREVENT Phase 3 Discrete HUD Pitch Tracking Task LAMARS Data

with Logarithmic Curve Fit ................................................................................... 3-13 3-11. HAVE PREVENT Phase 3 Target Tracking Task LAMARS Data with

Logarithmic Curve Fit............................................................................................ 3-13 3-12. HAVE OLOP Nichols Charts for Cases A, B and C........................................... 3-17 3-13. HAVE OLOP Phase 2 Sum-of-sines Task VISTA Data..................................... 3-20 3-14. HAVE OLOP Phase 3 Discrete HUD Pitch Tracking Task VISTA Data........... 3-20 3-15. HAVE OLOP Phase 2 Sum-of-sines Task VISTA Data with Logarithmic Fit... 3-23 3-16. HAVE OLOP Phase 3 Discrete HUD Pitch Tracking Task VISTA Data with

Logarithmic Fit ...................................................................................................... 3-23 4-1. MAX GAP (LAMARS) Nichols Charts for Cases B, N, W and Y......................... 4-3 4-2. MAX GAP (LAMARS) Phase 2 Sum-of-sines Task Data...................................... 4-6 4-3. MAX GAP (LAMARS) Phase 3 Discrete HUD Pitch Tracking Task Data ........... 4-6 4-4. MAX GAP (LAMARS) Phase 3 Target Tracking Task Data ................................. 4-7 4-5. MAX GAP (LAMARS) Phase 2 Sum-of-sines Task Data with Logarithmic Curve

Fit ............................................................................................................................. 4-9

xi

Figure .............................................................................................................................Page 4-6. MAX GAP (LAMARS ) Phase 3 Discrete HUD Pitch Tracking Task Data with

Logarithmic Fit ...................................................................................................... 4-10 4-7. MAX GAP (LAMARS) Phase 3 Target Tracking Task Data with Logarithmic

Fit ........................................................................................................................... 4-10 5-1. MAX GAP (VISTA) Nichols Charts for Cases B, N, W and Y.............................. 5-4 5-2. MAX GAP (VISTA) Phase 2 Sum-of-sines Task Data .......................................... 5-7 5-3. MAX GAP (VISTA) Phase 3 Discrete HUD Pitch Tracking Task Data ................ 5-7 5-4. MAX GAP (VISTA) Phase 2 Sum-of-sines Task Data with Logarithmic Curve

Fit ............................................................................................................................5--9 5-5. MAX GAP (VISTA) Phase 3 Discrete HUD Pitch Tracking Task Data with

Logarithmic Curve Fit............................................................................................ 5-10 6-1. LAMARS Combined Phase2 Sum-of-sines Dataset ............................................... 6-1 6-2. LAMARS Combined Phase 3 Discrete (HUD) Pitch-Tracking Dataset ................. 6-2 6-3. LAMARS Combined Phase3 Target Tracking Dataset........................................... 6-2 6-4. VISTA Combined Phase 2 Sum-of-sines Dataset ................................................... 6-3 6-5. VISTA Combined Phase 3 Discrete (HUD) Pitch-Tracking Dataset...................... 6-3 6-6. LAMARS and VISTA Combined Phase 2 Sum-of-sines Data ............................... 6-4 6-7. LAMARS and VISTA Combined Phase 3 Discrete Pitch-Tracking Data .............. 6-4 A-1. Augmented Dynamics SimulinkTM Model ............................................................ A-4 D-1. LAMARS Phase 2 Sum-of-sines Data, Case B, 15 deg/sec .................................. D-1 D-2. LAMARS Phase 2 Sum-of-sines Data, Case B, 30 deg/sec .................................. D-2 D-3. LAMARS Phase 2 Sum-of-sines Data, Case B, 60 deg/sec .................................. D-2 D-4. LAMARS Phase 2 Sum-of-sines Data, Case N, 15 deg/sec.................................. D-3 D-5. LAMARS Phase 2 Sum-of-sines Data, Case N, 30 deg/sec.................................. D-3

xii

Figure .............................................................................................................................Page D-6. LAMARS Phase 2 Sum-of-sines Data, Case N, 60 deg/sec.................................. D-4 D-7. LAMARS Phase 2 Sum-of-sines Data, Case W, 15 deg/sec................................. D-4 D-8. LAMARS Phase 2 Sum-of-sines Data, Case W, 30 deg/sec................................. D-5 D-9. LAMARS Phase 2 Sum-of-sines Data, Case W, 60 deg/sec................................. D-5 D-10. LAMARS Phase 2 Sum-of-sines Data, Case Y, 15 deg/sec................................ D-6 D-11. LAMARS Phase 2 Sum-of-sines Data, Case Y, 30 deg/sec................................ D-6 D-12. LAMARS Phase 2 Sum-of-sines Data, Case Y, 60 deg/sec................................ D-7 D-13. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case B, 15 deg/sec...... D-8 D-14. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case B, 30 deg/sec...... D-8 D-15. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case B, 60 deg/sec...... D-9 D-16. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case N, 15 deg/sec ..... D-9 D-17. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case N, 30 deg/sec ... D-10 D-18. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case N, 60 deg/sec ... D-10 D-19. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case W, 15 deg/sec... D-11 D-20. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case W, 30 deg/sec... D-11 D-21. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case W, 60 deg/sec... D-12 D-22. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 15 deg/sec ... D-12 D-23. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 30 deg/sec ... D-13 D-24. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 60 deg/sec ... D-13 D-25. LAMARS Phase 3 Target Tracking Data, Case B, 15 deg/sec ......................... D-14 D-26. LAMARS Phase 3 Target Tracking Data, Case B, 30 deg/sec ......................... D-14 D-27. LAMARS Phase 3 Target Tracking Data, Case B, 60 deg/sec ......................... D-15

xiii

Figure .............................................................................................................................Page D-28. LAMARS Phase 3 Target Tracking Data, Case N, 15 deg/sec ......................... D-15 D-29. LAMARS Phase 3 Target Tracking Data, Case N, 30 deg/sec ......................... D-16 D-30. LAMARS Phase 3 Target Tracking Data, Case N, 60 deg/sec ......................... D-16 D-31. LAMARS Phase 3 Target Tracking Data, Case W, 15 deg/sec ........................ D-17 D-32. LAMARS Phase 3 Target Tracking Data, Case W, 30 deg/sec ........................ D-17 D-33. LAMARS Phase 3 Target Tracking Data, Case W, 60 deg/sec ........................ D-18 D-34. LAMARS Phase 3 Target Tracking Data, Case Y, 15 deg/sec ......................... D-18 D-35. LAMARS Phase 3 Target Tracking Data, Case Y, 30 deg/sec ......................... D-19 D-36. LAMARS Phase 3 Target Tracking Data, Case Y, 60 deg/sec ......................... D-19 D-37. VISTA Phase 2 Sum-of-sines Data, Case B, 15 deg/sec................................... D-20 D-38. VISTA Phase 2 Sum-of-sines Data, Case B, 30 deg/sec................................... D-20 D-39. VISTA Phase 2 Sum-of-sines Data, Case B, 60 deg/sec................................... D-21 D-40. VISTA Phase 2 Sum-of-sines Data, Case N, 15 deg/sec................................... D-21 D-41. VISTA Phase 2 Sum-of-sines Data, Case N, 30 deg/sec................................... D-22 D-42. VISTA Phase 2 Sum-of-sines Data, Case N, 60 deg/sec................................... D-22 D-43. VISTA Phase 2 Sum-of-sines Data, Case W, 15 deg/sec.................................. D-23 D-44. VISTA Phase 2 Sum-of-sines Data, Case W, 30 deg/sec.................................. D-23 D-45. VISTA Phase 2 Sum-of-sines Data, Case W, 60 deg/sec.................................. D-24 D-46. VISTA Phase 2 Sum-of-sines Data, Case Y, 15 deg/sec................................... D-24 D-47. VISTA Phase 2 Sum-of-sines Data, Case Y, 30 deg/sec................................... D-25 D-48. VISTA Phase 2 Sum-of-sines Test Data, Case Y, 60 deg/sec........................... D-25 D-49. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case B, 15 deg/sec......... D-26

xiv

Figure .............................................................................................................................Page D-50. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case B, 30 deg/sec......... D-26 D-51. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case B, 30 deg/sec......... D-27 D-52. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case N, 15 deg/sec ........ D-27 D-53. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case N, 30 deg/sec ........ D-28 D-54. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case N, 60 deg/sec ........ D-28 D-55. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case W, 15 deg/sec ....... D-29 D-56. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case W, 30 deg/sec ....... D-29 D-57. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case W, 60 deg/sec ....... D-30 D-58. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 15 deg/sec ........ D-30 D-59. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 30 deg/sec ........ D-31 D-60. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 60 deg/sec ........ D-31

xv

List of Tables

Table Page 1-1. Summary of Famous Longitudinal PIO Events (McRuer, 1995:9)......................... 1-4 1-2. PIO Tendency Rating Scale (MIL-HDBK-1797, 1997:153) ................................ 1-10 2-1. Neal-Smith Pilot Models ....................................................................................... 2-14 3-1. HAVE PREVENT Case Characteristics (Hanley, 2003) ........................................ 3-3 3-2. HAVE PREVENT Pitch-to-Actuator Transfer Functions ( cG ) .............................. 3-4 3-3. HAVE PREVENT Neal-Smith Pilot Models .......................................................... 3-5 3-4. HAVE PREVENT Components for Cases A and B................................................ 3-6 3-5. Gap Criteria Values for HAVE PREVENT Cases A and B.................................... 3-7 3-6. Gap Criteria for HAVE PREVENT Case C ........................................................... 3-7 3-7. HAVE PREVENT Gap Criteria Summary............................................................. 3-8 3-8. HAVE PREVENT Correlation Confidence Levels............................................... 3-11 3-9. HAVE PREVENT Curve Fit Correlation Values.................................................. 3-11 3-10. HAVE OLOP Case Characteristics (Gilbreath, 2001) ........................................ 3-15 3-11. HAVE OLOP Pitch-to-Actuator Transfer Functions ( cG ).................................. 3-15 3-12. HAVE OLOP Neal-Smith Pilot Models.............................................................. 3-16 3-13. HAVE OLOP Components for Cases A and C ................................................... 3-17 3-14. Gap Criteria Values for HAVE OLOP Case A and C......................................... 3-18 3-15. HAVE OLOP Gap Criteria Summary ................................................................ 3-19 3-16. HAVE OLOP Correlation Confidence Levels .................................................... 3-21 3-17. HAVE OLOP Curve Fit Correlation Values ....................................................... 3-22 4-1. MAX GAP (LAMARS) Case Characteristics (Witte and others, 2003)................. 4-1

xvi

Table Page 4-2. MAX GAP (LAMARS) Pitch-to-Actuator Transfer Functions ( cG )...................... 4-2 4-3. MAX GAP (LAMARS) Neal-Smith Pilot Models.................................................. 4-3 4-4. MAX GAP (LAMARS) Components for Cases B, N, W and Y ............................ 4-4 4-5. Gap Criteria Values for MAX GAP (LAMARS) Cases B, N, W and Y................. 4-4 4-6. MAX GAP (LAMARS) Gap Criteria Summary .................................................... 4-5 4-7. MAX GAP (LAMARS) Correlation Confidence Levels ........................................ 4-8 4-8. MAX GAP (LAMARS) Curve Fit Correlation Values ........................................... 4-8 5-1. MAX GAP (VISTA) Case Characteristics (Witte and others, 2003)...................... 5-2 5-2. MAX GAP (VISTA) Pitch-to-Actuator Transfer Functions ( cG ) .......................... 5-2 5-3. MAX GAP (VISTA) Neal-Smith Pilot Models ...................................................... 5-3 5-4. MAX GAP (VISTA) Components for Cases B and W........................................... 5-4 5-5. Gap Criteria Values for MAX GAP (VISTA) Cases B and W ............................... 5-5 5-6. MAX GAP (VISTA) Components for Case N........................................................ 5-5 5-7. Gap Criteria Values for MAX GAP (VISTA) Case N ............................................ 5-5 5-8. Gap Criteria for MAX GAP (VISTA) Case Y ....................................................... 5-6 5-9. MAX GAP (VISTA) Gap Criteria Summary ......................................................... 5-6 5-10. MAX GAP (VISTA) Correlation Confidence Levels ........................................... 5-8 5-11. MAX GAP (VISTA) Curve Fit Correlation Values.............................................. 5-9 C-1. Minimum Values of the Correlation Coefficient for Confidence Level

(Wheeler and Ganji, 1996:147)............................................................................... C-2

xvii

List of Notations and Symbols

Symbol Definition Unit A Commanded Actuator Deflection deg Amax Maximum Actuator Deflection Available deg AFIT Air Force Institute of Technology - AFRL Air Force Research Laboratories - α, Alpha Angle-of-Attack deg dB Decibels - deg Degrees - δ Actuator Deflection Angle deg φ∆ Phase Angle rad

DoD Department of Defense - e, E Error - G(s) Generic Laplacian Transfer Function - Gactuator Actuator Transfer Function - Gaugmented Augmented Aircraft Transfer Function - GC Gap Criterion - Gc(s) Bare Aircraft Transfer Function - GP(s) Neal-Smith Pilot Model Transfer Function - HQDT Handling Qualities During Tracking - HUD Heads Up Display - jω Imaginary Laplacian Term -

xviii

Symbol Definition Unit k Feedback Gain Matrix - K* Intermediate Variable - Kp Neal-Smith Pilot Gain - Kα, Kalpha Angle-of-Attack Feedback Gain - Kq, Kq Pitch Rate Feedback Gain - LAMARS Large Amplitude Multimode Aerospace Simulator - Ln Natural Logarithm - MIL HDBK 1797 Military Handbook 1797 - n Number of Datapoints - N(A, ω) Describing Function - OLOP Open Loop Onset Point -

Phase Angle deg ω Frequency rad/sec ωa Actuator Gain - ωBW Band Width Frequency rad/sec ωsp Short Period Natural Frequency rad/sec PIO Pilot-induced Oscillations - PIOR Pilot-induced Oscillation Tendency Rating - q Aircraft Pitch Rate deg/sec R, rxy Correlation Factor - rad Radians - s Laplacian Operator -

xix

Symbol Definition Unit sec Seconds - θ, Theta Aircraft Pitch Angle deg T2 Time to Double Amplitude sec TLag Neal-Smith Lag Time Constant sec TLead Neal-Smith Lead Time Constant sec USAF TPS United States Air Force Test Pilot School - V True Velocity ft/sec VISTA Variable In-flight Stability Test Aircraft - VL Actuator Rate Limit deg/sec ζsp Short Period Damping Ratio -

1-1




I. Introduction

General

The purpose of this study was to investigate pilot-induced oscillations (PIO) and

determine a new method by which PIO tendency rating could be predicted. In particular,

longitudinal PIO in the presence of rate-limited actuators were singled out for

examination. The PIO rating scale used in this investigation was the PIO tendency

classification of the Department of Defense Interface Standard Flying Qualities of Piloted

Aircraft (MIL-HDBK-1797, 1997:152). While there are a number of PIO prediction

methods already published, this study will attempt a new approach.

This study was originated at the Air Force Institute of Technology (AFIT),

Wright-Patterson AFB, Ohio and was supported by the United States Air Force Test Pilot

School (USAF TPS). Research was conducted in both the Large Amplitude Multimode

Aerospace Simulator (LAMARS) at the Air Force Research Laboratory (AFRL), Wright-

Patterson AFB, Ohio and in the USAF NF-16D Variable In-flight Stability Test Aircraft

(VISTA) at Edwards AFB, CA. The VISTA aircraft is operated by USAF TPS and

supported by General Dynamics Advanced Information Systems of Buffalo, NY.

1-2

Background

Pilot-induced oscillations have been an aviation problem for over 100 years now.

The first incidence can be traced back to Wilbur and Orville Wright in 1903. When the

two brothers first took to the skies of Kitty Hawk, North Carolina, they experienced “a

mild longitudinal oscillation of the Wright Flyer” (Duda, 1995:288). The PIO problem

had just begun.

PIO Defined.

Before continuing with the century-long history of PIO, an understanding of the

term PIO is in order. A pilot-induced oscillation can be described as “an inadvertent,

sustained aircraft oscillation which is the consequence of an abnormal joint enterprise

between the aircraft and the pilot” (McRuer, 1995:2). Elaborated, a PIO is a complex

interaction between a pilot and his active involvement in an aircraft feedback system

(Klyde and others, 1995:14). The United State Department of Defense (DoD) defines

PIO as “sustained or uncontrollable oscillations resulting from the efforts of the pilot to

control the aircraft” (MIL-HDBK-1797, 1997:151). Although the key causal factor in

pilot-induced oscillation seems to be the pilot, it is important to make the assertion that,

generally, the pilot is not at fault and that there seems to be embedded in the flight control

system of the aircraft some tendencies predisposing the pilot-aircraft system toward PIO

occurrence (Klyde and others, 1995:14). In recent times, new terms have been put forth

to replace the familiar PIO such that the pilot’s guilt in such events is less likely to be

assumed. These include aircraft-pilot coupling (APC), pilot–in-the-loop oscillations and

pilot-assisted (or augmented) oscillations (Klyde and others, 1995:14). However,

experienced test pilots, including instructors at the US test pilot schools, and people in the

1-3

handling qualities community have expressed widespread allegiance to the traditional

term PIO and, therefore, this term will be used throughout this study (Mitchell and Hoh,

1995:16; Klyde and others, 1995:14).

In addition to defining what a PIO is, it is just as important to define what it is not.

A PIO could mean any oscillation that occurs during manual, piloted control. But some

of these situations could be the result of pilot overcontrol such as when a student pilot is

learning to land and balloons the aircraft. To an outsider, this could look like a PIO but

really is just part of standard pilot compensation lasting no more than one or two cycles

and is not a “real” PIO (Mitchell and Hoh 1995:17). Other researchers describe these as

“minor bobbles” that are often encountered as pilots get used to a new aircraft and is just

part of the learning experience (Klyde and others, 1995:14). It is also important to realize

that motions resulting from poor damping of the short period or dutch roll modes are not

PIO, when the motion does not result from the “efforts of the pilot to control the aircraft”

(Mitchell and Hoh, 1995:17-18).

To distinguish between these examples and a true PIO, some leading researchers

propose the following additional definition of PIO: “A PIO exists when the airplane

attitude, angular rate, or normal acceleration is 180 degrees out of phase with the pilot’s

control inputs” (Mitchell and Hoh, 1995:18). A great example of this phase lag can be

seen in Figure 1-1. This is the recorded data of the YF-22A accident which occurred on

25 April 1992 during a planned go-around at low altitude. This stripchart data depict a

180 degree phase difference between the aircraft pitch attitude and stick input.

1-4

Figure 1-1. YF-22A Accident Sequence (Hodgkinson, 1999:128)

PIO History.

The YF-22A PIO occurrence is just one of the most recent events. There is a long

history of PIO events in both operational and test flying as shown in Table 1-1.

Table 1-1. Summary of Famous Longitudinal PIO Events (McRuer, 1995:9)

Aircraft Type Summary of Incident XS-1 PIO during gliding approach and landing, 24 Oct 1947

XF-89A PIO during level off from dive recovery, early 1949 F-100 PIO during tight maneuvering X-15 Gliding flight approach and landing, 8 Jun 1959; Category II PIO

XF2Y-1 Post-takeoff destructive PIO YF-12 Mid-frequency severe PIO; Category III PIO

Space Shuttle ALT-5 during landing approach glide, 26 Oct 1977; Category II PIO DFBW F-8 PIO during touch and goes, 18 Apr 1978; Category III PIO

YF-22 PIO after touchdown and wave off in afterburner, 25 Apr 1992 JAS 39 PIO during approach, 1990; 1993; Category II – III PIO MD-11 China Eastern Airlines Flt 583, 6 Apr 1993; Inadvertant slat deployment

F-4 Low altitude record run second pass, 18 May 1961; Destructive PIO

1-5

The onset of PIO occurs when the pilot attempts tight control. The DoD defines

Category A Flight Phases as “those nonterminal Flight Phases that require rapid

maneuvering, precision tracking, or precise flight path control” (MIL-HDBK-1797,

1997:80). Types of maneuvers included in this category are in-flight refueling (receiver),

air-to-air combat, and formation flying (MIL-HDBK-1797, 1997:80). Category C Flight

Phases also require “accurate flight-path control” and include takeoffs, approaches and

landings (MIL-HDBK-1797, 1997:80). Almost all of the events listed in Table 1-1 fall

into Category A or Category C.

PIO can range in severity from benign to catastrophic (Anderson and Page,

1995:278). Benign PIO may not lead to loss of aircraft or life, but they can affect

operational missions. For example, receiver air-refueling may take longer than expected

causing delays in mission progress (Anderson and Page, 1995:278). This is still far easier

to accept than the more severe extreme in which aircraft and pilots have both been lost

due to PIO. On 18 May 1961, US Navy pilot Cmdr Jack Feldman and his F-4B aircraft

were lost when a PIO disintegrated his aircraft during the second pass of a low altitude

speed record attempt due to PIO related to low short-period damping (McCruer, 1995:9).

Even worse, the potential for PIO has actually increased due to the evolving use

of highly augmented, fly-by-wire aircraft (Liebst and others, 2002:740). Development

flight test of the Boeing 777, C-17A and Saab JAS 39 Gripen bear this out. The 777, a

fly-by-wire transport, encountered a PIO situation during a Flaps 20 landing. The

spoilers automatically deployed upon touchdown and the pilot countered with forward

column motion resulting in a PIO that lasted for “about 3 cycles” (Dornheim, 1995:32).

C-17 longitudinal PIO tendencies were discovered during dutch roll testing, air-refueling

1-6

breakaways and three-engine landings. Additionally, further PIO tendencies were

exposed during handling qualities during tracking (HQDT) exercises with an A-37 target

aircraft (Preston and others, 1996:20-49). The discovery of PIO tendencies in the JAS 39

was not so uneventful. Two aircraft were lost during different stages of aircraft

development. The first accident occurred in 1989 with the test aircraft having had only

5.3 hours of flight time when, during a landing attempt, it experienced “roll pendulums”

at low altitude and later “pendular movements in pitch, also” at an altitude of

approximately 10 to 15 meters. The mishap pilot attempted a go-around but the aircraft

“hit the runway with the left wingtip and main gear as well as the nose portion” (Ahlgren,

1989:12). The pilot survived and returned to test flying the JAS 39. In 1993 while

performing in an airshow in Stockholm, Sweden, he experienced a lateral PIO

compounded with an uncommanded pitch up to high angle-of-attack. The aircraft

diverged from controlled flight but the pilot ejected safely (Jensen and others,

undated:12).

PIO Categories.

PIO can be divided into three categories (McRuer, 1995:79-80). Category I PIO

are essentially linear pilot-vehicle system oscillations; they are usually the low frequency

consequence of excessive high frequency lag in the aircraft’s linear dynamics (Klyde and

others, 1996:17). On the other hand, Category III PIO are associated with essentially

nonlinear pilot-vehicle systems with transitions; they are the result of abrupt shifts in

either the effective controlled-element dynamics or in the pilot’s behavioral dynamics

(Klyde and others, 1996:17). Category II PIO are defined as quasi-linear pilot-vehicle

system oscillations associated with control surface rate or position limiting (McRuer,

1-7

1995:80). In other words, rate-lmited actuators. They represent a transition from linear

dynamics to nonlinear effects and were the focus of this investigation.

PIO Category II.

One reason for focusing on Category II PIO is that “almost all of the severe PIO

time history records available for operational and flight test aircraft … show surface rate

limiting …in the fully developed oscillation” (Klyde and others, 1996:15). Other

researchers have similar opinions: “It seems to be true that all recent PIO have exhibited

rate saturation” (Duda , 1995:289). A prime example is the development of the C-17 in

which “rate limiting was involved in all the subject [PIO] events and is viewed to be the

primary cause of the longitudinal PIO” (Preston and others, 1996:20).

There are two major detrimental effects of rate limiting. One is that it adds to the

effective lag in series with the pilot therefore making the effective aircraft dynamics

worse and the other is that it exposes the bare aircraft dynamics in stability augmented

aircraft. On the positive side, however, rate limiting tends to confine the ultimate

amplitude of the pilot-vehicle system oscillation (Klyde and others, 1996:15). Figure 1-2

shows the differences in the output of a typical first-order actuator as the effects of

increasing amplitude drive the system toward rate limiting. The figure displays the

characteristic triangular wave pattern of the actuator position reversing when equal to the

commanded position. The “boxcar-like” rate response is clipped at 40 deg/sec in this

example and is also a trait of a rate-limited actuator (Klyde and others, 1996:25). The

output was generated using the MatlabTM/SimulinkTM software model which is shown in

Figure 1-3 (MatlabTM/SimulinkTM, 2001).

1-8

Figure 1-2. Actuator Saturation Example Using an Input = A sin(ωi t)

Figure 1-3. SimulinkTM Actuator Model

PIO Scales.

Before discussing PIO prediction, a scale must be selected which distinguishes

increasing severity of PIO. Figure 1-4 shows just such a scale, the PIO tendency

classification (MIL-HDBK-1797, 1997:152). This scale is in common use among test

1-9

pilots to rate the susceptibility of aircraft to PIO and a large amount of historical data is

available based on this scale. Comparing this scale to Cooper-Harper Handling Qualities

Levels, a PIO rating of 1 or 2 is approximately equivalent to a Level 1 aircraft, a PIO

rating of 3 or 4 is similar to a Level 2 aircraft, a PIO rating of 5 is Level 3 and a PIO

rating of 6 is extremely dangerous (MIL-HDBK-1797, 1997:151). These PIO ratings are

more fully described in the PIO tendency rating scale of Table 1-2 (MIL-HDBK-1797,

1997:322).

Figure 1-4. PIO Tendency Classification (MIL-HDBK-1797, 1997:152)

1-10

Table 1-2. PIO Tendency Rating Scale (MIL-HDBK-1797, 1997:153)

PIO Prediction Methods.

Using this scale, several criteria for PIO prediction have been developed for

Category I and Category II PIO (Mitchell and Klyde, 1998:417-426). Category I PIO

prediction methods include the Bandwidth/Pitch Rate Overshoot Method, the original

Neal-Smith criteria, the Neal-Smith criteria as modified by the Moscow Aviation

Institute, Smith-Geddes PIO criteria, and the Gibson criteria (Mitchell and Klyde,

1998:417-426). A thorough examination of these methods led Mitchell and Klyde to

conclude that “we are in reasonably good shape in predicting Category I PIO” (Mitchell

and Klyde, 1998:417-426). Satisfactory development of Category II PIO prediction

methods, on the other hand, is not as complete.

DESCRIPTION NUMERICALRATING

No tendency for pilot to induce undesirable motions. 1 Undesirable motions tend to occur when pilot initiates abrupt maneuvers or attempts tight control. These motions can be prevented or eliminated by pilot technique.

2

Undesirable motions easily induced when pilot initiates abrupt maneuvers or attempts tight control. These motions can be prevented or eliminated but only at sacrifice to task performance or through considerable pilot attention and effort.

3

Oscillations tend to develop when pilot initiates abrupt maneuvers or attempts tight control. Pilot must reduce gain or abandon task to recover.

4

Divergent oscillations tend to develop when pilot initiates abrupt maneuvers or attempts tight control. Pilot must open loop by releasing or freezing the stick.

5

Disturbance or normal pilot control may cause divergent oscillation. Pilot must open control loop by releasing or freezing the stick. 6

1-11

Some work in Category II prediction methods has been accomplished or is in

progress. These efforts include modifying the Bandwidth/Phase Delay criteria to

Category II cases, the Time-domain Neal-Smith criteria, the Open Loop Onset Point

(OLOP) criteria and a power spectral density method using a structural model of the

human pilot (Mitchell and Klyde, 1998:417-426). These efforts continue despite the

assertion by Anderson and Page that “…the adaptive nature of the pilot makes such

oscillations [PIO] difficult to predict” (Anderson and Page, 1995:278).

Gap Criterion.

The Department of Aeronautical and Astronautical Engineering at the Air Force

Institute of Technology (AFIT) now proposes a new criterion for predicting PIO ratings

dubbed the Gap Criterion. It is based on describing function techniques, modified Neal-

Smith pilot models and actuator input amplitude ratios. The goal of this study was to

determine if specific relationships between Gap Criterion and PIO tendency rating for

Category II PIO due to rate-limited actuators exists and, if so, to what extent.

Objectives

The primary objective of this study was to develop a new criterion for predicting

PIO tendency rating by:

1) Exploiting previously defined describing function methods for determining

Category II PIO characteristics based on rate-limited actuators.

2) Defining Gap Criterion.

3) Applying the new method to existing historical data from similar test programs.

4) Conducting both simulator and flight tests to expand this database.

5) Making recommendations on the implementation of this new criterion.

1-12

Approach

The following steps were taken for this project:

1) The Gap Criterion was applied to data from previous simulator and flight test

projects in which rate limiting PIO effects were studied. These included USAF

TPS projects HAVE OLOP and HAVE PREVENT (Gilbreath, 2001; Hanley,

2003). The Gap Criterion was calculated for each test case of bare aircraft

dynamics and rate-limit. These Gap Criteria were then matched with their

respective PIO tendency ratings and plotted in pairs. Correlation probability

confidence level was then determined along with curve fits.

2) Based on the observed data from these studies, a broader range of longitudinal

flight control system dynamics with varying short period characteristics were

chosen to augment the database. These configurations were tested in both the

LAMARS simulator and the VISTA NF-16D aircraft.

Scope

This research project was limited in scope and constrained in certain areas:

1. The PIO investigated were strictly longitudinal, Category II PIO due to rate

limiting of the actuator.

2. Only three rate limits were chosen: 15 deg/sec, 30 deg/sec and 60 deg/sec.

3. Only four distinct bare aircraft dynamics cases were tested.

4. Tracking tasks were created with HUD generated and target aircraft sorties. No

offset landing tasks were planned.

5. Due to schedule and budget, the simulator test portion was limited to two days and

three pilots. The flight test portion was limited to eight sorties and 10.8 hour

2-1

II. Theory

This chapter will discuss describing functions and how they can be used to

understand and predict PIO onset when considered in the context of rate-limited

actuators. Further, the Neal-Smith pilot model will be explained followed by an example

integrating all of these concepts. The basis of the Gap Criterion will then be covered.

Describing Function Development

Observing the time history of the YF-22 PIO from Figure 1-1, it can be seen that

the input is approximately sinusoidal. This is true in general of all PIO incidents (Klyde

and others, 1996:37). The describing function technique can be used for limit cycle

analysis due to the fact that that the form of the signals in a limit-cycling system, such as

a PIO, is usually approximately sinusoidal (Slotine and Li, 1991:157).

Any system which can be rearranged into the form shown in Figure 2-1, where w

and G(p) represent nonlinear and linear elements, respectively, can be studied using

describing functions (Slotine and Li, 1991:162). Examples of nonlinear elements include

dead-zones, hysteresis or rate saturations. Rate saturations were the focus of this study.

w=f(x) G(p) + - y(t) r(t) = 0 x(t) w(t)

Figure 2-1. Example of a Nonlinear System

For the basic version of the describing function method, the system has to satisfy

the following four conditions (Slotine and Li, 1991:164):

2-2

1) There is only a single nonlinear component

2) The nonlinear component is time-invariant

3) Corresponding to a sinusoidal input sin( )x A tω= , only the fundamental

component w1(t) in the output w(t) has to be considered

4) The nonlinearity is odd

Consider a sinusoidal input of the form sin( )x A tω= entering the nonlinear

element of the system shown in Figure 2-1. Due to nonlinear effects, the output, w(t), is

“often a periodic though non-sinusoidal function” (Slotine and Li, 1991:165). The output

function w(t) can be expanded using Fourier series as seen in Equation 1 and the

succeeding derivation (Slotine and Li, 1991:165):

0

1( ) [ cos( ) sin( )]

2 n nn

aw t a n t b n tω ω∞

=

= + +∑ (1)

where

01 ( ) ( )a w t d t

π

π

ωπ −

= ∫ (2)

1 ( )cos( ) ( )na w t n t d tπ

π

ω ωπ −

= ∫ (3)

1 ( )sin( ) ( )nb w t n t d tπ

π

ω ωπ −

= ∫ (4)

Applying condition four above, for all odd functions 0 0a = (Slotine and Li,

1991:166). Further, applying the third assumption means discarding all other terms

except 1n = (Slotine and Li, 1991:166). This leaves:

1 1 1( ) ( ) cos( ) sin( )w t w t a t b tω ω≈ = + (5)

2-3

which can be rewritten as

1( ) sin( )w t M tω φ= + (6)

where

2 21 1M a b= + (7)

1 1

1

tan ab

φ − =

(8)

Rewritten in complex notation leads to:

( ) ( )1 1 1( ) ( )j t j tw t Me b ja eω φ ω+= = + (9)

Finally, the describing function, ( , )N A ω , is defined to be the complex ratio of

the fundamental component of the nonlinear element to the input sinusoid. This is shown

in Equation 10:

( )( )

1 1( )

1( , )j t

jj t

Me MN A e b jaAe A A

ω φφ

ωω+

= = = + (10)

Saturation Nonlinearity Describing Function

Now consider the saturation input-output relationship shown in Figure 2-2 below:

γ ka

ka 0 ωt

w(t) w

0

k

ωt

γ π/2

A x(t) 0

a x

Figure 2-2. Saturation Nonlinearity and the Corresponding Input-Output Relationship

(Slotine and Li, 1995:173)

2-4

From the figure, it is apparent that if our input, ( ) sin( )x t A tω= , has a maximum

amplitude A ≤ a then the input remains linear and the output is just w(t) = kAsin(ωt). But

if the maximum amplitude, A, is greater than a, clipping occurs and the value of w(t) can

be split into two sets over the first quarter of the symmetric output (Slotine and Li,

1995:173):

sin( )

( )kA t

w tka

ω=

0

2

t

t

ω γπγ ω

≤ ≤

< ≤ (11)

where γ = sin-1(a/A)

The output w(t) is an odd function, implying 1 0a = in Equation 5. Further,

dividing the output into four quarters yields a new equation for b1

2

10

4 ( )sin( ) ( )b w t t d tπ

ω ωπ

= ∫ (12)

2

21

0

4 4sin ( ) ( ) sin( ) ( )b kA t d t ka t d tπγ

γ

ω ω ω ωπ π

= +∫ ∫ (13)

2

1 2

2 1kA a abA A

γπ

= + −

(14)

Substituting 1 0a = and Equation 14 into Equation 10 leaves (Slotine and Li, 1995:174):

2

112

2( , ) sin 1b k a a aN AA A A A

ωπ

−

= = + −

(15)

Closed Loop Describing Function Approximation

Now, consider the block diagram in Figure 2-3 of a first order actuator system and

the derivations which follow (Klyde and others, 1996:36-46).

2-5

eδ

e s1eδ eδe

+−

LV

Le

aω

LV-

1

commandeδ

Figure 2-3. Actuator Model Development (Klyde and others, 1995:22)

The nonlinear portion of this model is exactly the same as the saturation

nonlinearity discussed previously. Substituting the appropriate new nomenclature and

letting ( ) sin( )e t E tω φ= + replace x(t), leads to the following describing function for the

nonlinear element:

2

12

2( , ) sin 1a L L Le e eN AE E E

ωωπ

− = + −

(16)

Further, by using series expansions for both the arcsine term and the square root, the

describing function can be approximated by:

3 2

2

2 1 1( , ) 16 2

a L L L Le e e eN AE E E E

ωωπ

= + + ⋅⋅⋅ + − − ⋅⋅⋅ (17)

Keeping only the first order linear terms yields:

2 4( , ) a aL L Le e eN AE E E

ω ωωπ π

= + = (18)

Substituting L a LV eω= leads to:

4( , ) LVN AE

ωπ

= (19)

2-6

Next, consider the revised block diagram shown in Figure 2-4 and determine the

closed loop transfer function, treating N as a constant.

N 1s + -

δ (s) e

δ (s) e Command e(s)

Figure 2-4. Closed Loop Actuator Transfer Function Diagram

Treating N as a constant and utilizing linear block diagram transfer function

techniques, the relationship of e(s) to ( )Commande sδ is:

( ) 1( ) 1Commande

e sNss

δ=

+ (20)

Assuming ( ) sin( )Commande t A tδ ω= and ( ) sin( )e t E tω φ= + and substituting jω for s,

the equation for the magnitude of this transfer function becomes:

2

2

( ) 1 sin( )( ) sin( )

1Commande

e s E t Es A t AN

ω φδ ω

ω

+= = =

+

(21)

Rearranging Equation 19 in terms of E gives:

4 LVENπ

= (22)

Substituting Equation 22 into Equation 21 and rearranging terms yields:

2

14 L

NAV

ω

π ω=

−

(23)

2-7

Now, still treating N as a constant and utilizing standard block diagram transfer

function techniques, the relationship of ( )e sδ to ( )Commande sδ is:

( ) 1( ) 1Command

e

e

s Nss s NN

δδ

= =++

(24)

and its magnitude is

2

2 2

( )( )

Command

e

e

j Nj N

δ ωδ ω ω

=+

(25)

and substituting Equation 23 into Equation 25 gives

( ) 4( )

Command

e L

e

j Vj A

δ ωδ ω π ω

= (26)

Solving for the phase angle of Equation 24 yields:

1( ) tan( )

Command

e

e

jj N

δ ω ωδ ω

− − =

(27)

and substituting Equation 23 yields

2

1( ) tan 1( ) 4

Command

e

e L

j Aj V

δ ω π ωδ ω

− = − −

(28)

Sinusoidal Input/Triangle Output Describing Function Approximation

Another describing function approximation can be made by utilizing the observed

characteristics of a saturated actuator. The input, xi(t), is sinusoidal in nature and the

output, x0(t), takes on the familiar saw tooth triangle shape as shown in Fig 2-5 (Klyde

and others, 1995:42-46):

2-8

t0

t i

x 0

x i x i (t)

tD

x0(t)

Rate LimitingElement

x0(t) x i (t)

Figure 2-5. Rate-Limiting Input and Output (Klyde and others, 1995:42-46)

As before, let the input be sinusoidal as shown in Equation 29:

max

( ) sin( )i ix t x tω= (29)

and the derivative or input rate is:

max

( ) cos( )i ix t x tω ω= (30)

Now, let 2 Tω π= where 4 iT t= . Then the maximum input rate is

max

max 2i

ii

xx

tπ

= (31)

The rate of the output, 0x , is equal to the slope of the output and is given by

00

0

xxt

= ± (32)

Now, take the relationship of the output rate to the input rate in the range of t0 and

solve for the ratio of output to input magnitude as:

2-9

max

maxmax

0 0 0

0 0

22

i i

i ii

xx x x tt t t xx

ππ

= =

(33)

Recognizing t0 equals ti, rearranging terms and introducing a new variable *K gives:

max max

0 0 *2i i

x x Kx x

π= = (34)

Rewriting this expression in terms of the Figure 2-3 variables and recognizing that the

output rate when saturated is VL and the maximum input rate is Aω leaves

*2

LVKA

πω

= (35)

The describing function magnitude is then expressed using the *K value multiplied by

the Fourier fundamental of the triangle wave as seen in Equation 36 (Klyde and others,

1996:45).

2

( ) 8 4*( )

Command

e L

e

j VKj A

δ ωδ ω π π ω

= = (36)

This is exactly the same expression derived earlier for the closed loop actuator describing

function magnitude. To obtain the phase angle of the input/output relationship, the term

tD as shown in Figure 2-5 must be determined. The input and output amplitudes are equal

when i Dt t t= + .

( )max 0sini i Dx t t xω + = (37)

Simplifying this expression by substituting max0* iK x x= , expanding sin[ω(ti + tD)], and

substituting 2itω π= results in (Klyde and others, 1996:45):

cos( ) *Kφ∆ = (38)

2-10

where Dtφ ω∆ = is the phase angle between the input and output. Solving for φ∆ and

noting that it is a phase lag gives Equation 39:

2

1 1( ) 1cos ( *) tan 1( ) *

Command

e

e

jKj K

δ ωφδ ω

− − −∆ = − = = − −

(39)

Now to compare with the closed-loop describing function phase angle, substitute

( )( )* 2 LK V Aπ ω= into Equation 39, which results in

2

1( ) 2tan 1( )

Command

e

e L

j Aj V

δ ω ωδ ω π

− = − −

(40)

This is slightly different from the closed loop describing function phase angle expression.

These phase angle differences are shown as a function of *K in Figure 2-6:

Figure 2-6. Describing Function Phase Angle Comparison

2-11

These two describing function approximations were introduced to show that

similar results can be derived from different methods. According to Klyde, the more

accurate of these two describing function approximations for application to Category II

PIO is the sinusoidal input/triangle output solution (Klyde and others, 1996:46).

Therefore, this describing function will be used throughout the remainder of this study.

Applying Describing Function Results to Predict PIO

Consider the longitudinal closed-loop system shown in Figure 2-7. ( )pG s

represents a model of the pilot and Gc (s) represents a model of the bare aircraft

dynamics. The remaining elements are equivalent to the rate-limited actuator model

previously discussed in Figure 2-3.

G p (s) G c (s) 1 s

e(s) θ Command (s) θ Error (s) θ (s) δe(s)δ e Command (s) δe(s)

.VL

-VL

ωa

k

Figure 2-7. Pitch Tracking Closed Loop System

The linear elements ( )pG s and ( )cG s can be combined into one linear element,

( )G s and the nonlinear element, ( , )N A ω , remains separate as shown in Figure 2-8.

N(A,ω) G(s)

+ -

θ Command (s)

θ Error (s)

θ (s)

Figure 2-8. Simplified Pitch Tracking Closed Loop System

2-12

This model can then be applied to a limit cycle analysis. The requirement for a

neutrally damped oscillation is simply that the open-loop amplitude ratio be equal to 1.0

and the phase be -180º (Klyde and others, 1996:54). In order for a PIO to persist, the

system shown in Fig 2-8 must satisfy the Nyquist criteria shown in the following

equation (Klyde and others, 1996:54):

( ) ( , ) 1G j N Aω ω = − (41)

or rearranged

1( )( , )

G jN A

ωω−

= (42)

The easiest way to view the application of this equation is to plot the open-loop

magnitude and phase values of the negative inverse describing function, 1 ( , )N j Aω− ,

using the *K solutions from Equations 36 and 39 as well as the open-loop magnitude

and phase of ( )G jω . If the two plots intersect, a PIO is predicted (Klyde and others,

1996:63). This will be shown by means of an example later in this chapter. The *K

solutions for the negative inverse describing function are shown below in Equations 43

and 44 (Liebst, 2002):

10 2

1 8 *20( *)

KLogN K π− = −

(dB) (43)

( )11 180 cos * 180( *)

KN K π

−−= − (deg) (44)

2-13

Pilot Model

There are many pilot models to choose from in the literature. Some believe that a

simple gain with no phase lag best represents the pilot in the PIO situation (Klyde and

others, 1996:54). Others believe structural models are better predictors (Mitchell and

Klyde, 1998:426). In another recent study, the Neal-Smith pilot model was judged to

best represent the pilot model prior to the onset of rate limiting (Gilbreath, 2001:7-3).

Therefore, in this study, the Neal-Smith pilot model will be utilized.

The Neal-Smith pilot model is useful for pilot-aircraft pitch attitude control loops

with unity-feedback and has the following characteristics (MIL-HDBK-1797, 1997:237):

1. Adjustable gain

2. Time delay

3. Ability to develop lead, or to operate on derivative or rate information

4. Ability to develop lag, or to smooth inputs

5. Ability to provide low-frequency integration

The Neal-Smith pilot model can take on one of two forms. This determination is

based on the whether constant speed or two-degree-of-freedom equations are used to

represent the bare aircraft dynamics. These are typified by noting whether or not a free

integrator is contained in the denominator of the aircraft pitch transfer function.

Otherwise, three-degree-of-freedom equations or flight control system utilizing attitude

stabilization will require a different form. Table 2-1 shows these two transfer functions

for the Neal-Smith pilot model (MIL-HDBK-1797, 1997:237; Bailey and BidLack,

1995:8).

2-14

Table 2-1. Neal-Smith Pilot Models Aircraft Transfer Function

with a Free Integrator Aircraft Transfer Function without a Free Integrator

( )( )

0.251( )

1Lead s

p pLag

T sG s K e

T s−+

=+

( ) ( )

( )0.255 1 1

( )1

Lead sp p

Lag

s T sG s K e

s T s−+ +

=+

The theory states that the pilot chooses his gain, pK , and his lead/lag time

constants, TLead and TLag, to attain a certain bandwidth. This bandwidth varies with the

flight phase category. For example, for Category A flight phase maneuvers such as air-

to-air dogfighting, the required bandwidth is 3.5 rad/sec. This is measured at a closed-

loop phase of –90 degrees. Further, the pilot adjusts pK , TLead and TLag to minimize droop

to no greater than 3 dB for Level 1 performance and no greater than 9 dB for Level 2

over the frequency range from 0 to 10 rad/sec while at the same time minimizing closed

loop resonance (MIL-HDBK-1797, 1997:239). The phase lag term, e-0.25s, represents

delays in the pilot’s neuromuscular system (MIL-HDBK-1797, 1997:239). A graphical

depiction of these pilot efforts can be seen in Figure 2-9.

2-15

Figure 2-9. Neal-Smith Pilot Model Constraints

Max Droop < 3 dB for Level 1

Minimize Resonance

Closed Loop Bode Diagram

( )( )

ssCommand

θθ

( )( )

ssCommand

θθ

BWω

2-16

Gap Criterion

Utilizing the previous theoretical developments, a systematic process relating

aircraft plant dynamics and actuator rate limits to PIO tendency rating will be introduced.

The procedure is called the Gap Criterion and is based on the block diagram shown in

Figure 2-10:

G p (s) G c (s) 1 s

e(s) θ Command (s) θ Error (s) θ (s) δe(s)δ e Command (s) δe(s)

.VL

-VL

ωa

k

*actuator augmentedG G

Figure 2-10. Pitch Tracking Closed-Loop System for Gap Criterion

In modern fly-by-wire aircraft, feedback is an integral part of obtaining more

desirable closed loop flying qualities. However, as mentioned earlier, rate limiting

exposes the unaugmented dynamics and adds phase lag (Hanley, 2003:1-3). A pilot

suddenly faced with different flying qualities will not be able to adjust his gain, lead or

lag properties instantaneously. He will therefore continue to fly in such a manner as if

the augmented aircraft dynamics were still in place. Therefore, the term G(jω) from

Equation 42 is the product of the bare aircraft dynamics, Gc(s), convolved with the Neal-

Smith pilot model, Gp(s). This idea is incorporated in the derivation of the Gap

Criterion.

When rate limiting is not occurring, the actuator dynamics from Figure 2-10 can

be determined from block diagram methods and is Gactuator = ( )a asω ω+ .

2-17

Gap Criterion Formulation.

Computing the Gap Criterion consists of the following steps:

1. Determine the bare aircraft pitch-to-actuator transfer function, ( )( )( )c

e

sG ss

θδ

= .

2. If the short period poles of ( )cG s are unstable then the Gap Criterion

automatically equals zero. This is due to control amplitudes approaching

zero which cause an immediate departure from controlled flight due to

dynamic instability resulting from actuator rate saturation.

3. Determine actuator dynamics for the following form: aactuator

a

Gsωω

=+

.

Typically, 20aω = and this will be used throughout this study (Liebst,2001).

4. Determine an appropriate optimized Neal-Smith pilot model, ( )pG s , for the

augmented aircraft dynamics plus actuator dynamics, *actuator augmentedG G .

5. Plot the open-loop magnitude and phase of the bare aircraft dynamics

convolved with the Neal-Smith pilot model dynamics, ( )* ( )c pG s G s on a

Nichols chart

6. Plot the negative inverse describing function open-loop magnitude and phase

on the same Nichols chart using *K . See Equations 43 and 44.

7. Determine the resulting type by reference to Figure 2-11 and then compute

the Gap Criterion by following the steps of that type.

2-18

Type I Type II

Type III Type IV

Figure 2-11. Four Resulting Gap Criterion Types

2-19

Type I.

1-1. Determine the minimum amount by which the pilot would need to

effectively increase gain, pK∆ (dB), such that the two magnitude-phase

lines just intersect at a frequency greater than the -3 dB Neal-Smith

maximum droop frequency as shown in Figure 2-12:

Figure 2-12. Case I Effective Pilot Gain Increase

1-2. Determine the values of K* and ω (rad/sec) at this intersection

1-3. Determine the commanded actuator deflection amplitude, A, utilizing the

following equation where VL is the known actuator rate limit in deg/sec:

2 *LVA

Kπω

=

1-4. Normalize this amplitude by dividing by the maximum available actuator

deflection, Amax

1-5. The Gap Criterion is this normalized amplitude multiplied by ∆Kp:

( / 20)

max

*10 pKAGap CriterionA

∆=

K p∆

2-20

Type II.

2-1. Determine the minimum amount by which the pilot would need to

effectively decrease gain, pK∆ (dB), such that the two magnitude-phase

lines only intersect in one place at a frequency greater than the -3 dB Neal-

Smith maximum droop frequency as shown in Figure 2-13:

Figure 2-13. Case II Effective Pilot Gain Decrease

2-2. Determine the values of K* and ω (rad/sec) at this intersection



2 *LVA

Kπω

=


deflection, Amax

2-5. The Gap Criterion is this normalized amplitude multiplied by ∆Kp.

( / 20)

max

*10 pKAGap CriterionA

∆=

K p∆

2-21

Type III.

3-1. Determine the values of K* and ω (rad/sec) at the intersection



2 *LVA

Kπω

=


deflection, Amax

3-4. The Gap Criterion is this normalized amplitude.

max

AGap CriterionA

=

Type IV.

4-1. No determination of Gap Criterion can be made.

Example of Gap Criterion Application

Reconsider the closed loop system of Figure 2-10 with the following characteristics:

• ( )( )( )2

4.5 1.5( )

3 6c

sG s

s s s+

=+ +

• 2020actuatorG

s=

+

• k = 0 (unaugmented), therefore augmented cG G=

• VL = 30 deg/sec

• Maximum actuator deflection: max

30 degeδ =

• Category A flight phase, Neal-Smith bandwidth = 3.5 rad/sec

2-22

Utilizing the USAF Wright Laboratory Flight Dynamics Directorate’s MATLABTM

Interactive Flying Qualities Toolbox for Matlab (Domon and Foringer, 1996), the Neal-

Smith pilot model was found to be:

0.25sp e

1)(0.0001s1)(0.583s0.856(s)G −

++

=

The open-loop magnitude and phase of *actuator augmentedG G are plotted in Figure 2-14 as

well as the open-loop magnitude and phase of the negative inverse describing function. It

can be seen that an open-loop gain increase (∆Kp) of 7.502 dB is required for the two

lines to meet. At this intersection, the values for K* and ω are 0.7635 and 3.9418 rad/sec,

respectively. After calculating the amplitude, A = 15.66 deg, the result is normalized by

dividing by 30 deg (δe max). This normalized result is multiplied by ∆Kp to yield the Gap

Criterion. In this example the Gap Criterion equals 1.238.

This example showed how preflight Gap Criterion can be calculated. It is

expected that matching these values with their respective PIO Tendency Ratings should

yield some correlation. This will be accomplished by examining historical databases and

then collecting additional data in the LAMARS simulator and VISTA aircraft.

Figure 2-14. Nichols Chart of the Example Problem

∆Kp = 7.502 dB

K* = 0.7635 ω = 3.9418 rad/sec

3-1

III. Analysis of Selected Historical Data

In this section, two previous PIO studies, HAVE PREVENT and HAVE OLOP,

will be examined using the Gap Criterion (Hanley, 2002; Gilbreath, 2000).

HAVE PREVENT Analysis

HAVE PREVENT was a simulator and flight test study comparing two different

PIO prevention filters. The study was conducted as part of a Test Management Project at

the USAF Test Pilot School and a thesis sponsored by the Air Force Institute of

Technology (Hanley and others, 2002; Hanley, 2003). The two PIO filters examined

were the Feedback with Bypass and the Derivative Switching filters. In order to establish

a baseline, runs with neither filter engaged were conducted in both the LAMARS

simulator and VISTA NF-16D aircraft. However, few no-filter data points were collected

during the flight test portion. Therefore, only the simulator data will be examined.

In this simulator, pilot-induced oscillation tendency ratings (PIOR) were gathered

in two distinct phases with specific piloting tasks and in which the evaluation pilot was

blind to the randomized bare aircraft dynamics and rate limit combinations. In Handling

Qualities Phase 2 testing, a precision aimpoint was tracked as “aggressively and

assiduously as possible, always striving to correct even the smallest errors” using a

piloting technique known as Handling Qualities During Tracking (HQDT) (Brown and

others, 2002:21-18). The precision aimpoint involved a sum-of-sines head’s up display

(HUD) tracking task, as shown in Figure 3-1 (MIL-HDBK-1797, 1997:108m). Handling

Qualities Phase 3 “operational” testing was also accomplished and involved two different

tasks (Brown and others, 2002:21-19). These tasks were the discrete HUD pitch tracking

task and computer generated aircraft target tracking task shown in Figures 3-2 and 3-3.

3-2

25

35

300

14.5

15.5

15,000

05 060 07

5 5

5 5 MOVING COMMAND BAR

FIXED AIRCRAFT SYMBOL

10 mil FIXED RETICLE


Figure 3-1. Phase 2 Sum-of-sines Pitch Tracking Task

25

35

300

14.5

15.5

15,000

05 060 07

5 5

5 5 MOVING COMMAND BAR




Figure 3-2. Phase 3 Discrete HUD Pitch Tracking Task

3-3

25

35

300

14.5

15.5

15,000

05 060 07

5 5

5 5




Target

25

35

300

14.5

15.5

15,000

05 060 07

5 5

5 5




Target

Figure 3-3. Phase 3 Target Tracking Task

The study involved four test cases labeled A through D. Each test case had

different bare aircraft dynamics but all were augmented with angle-of-attack and pitch-

rate feedback to produce almost identical closed loop dynamics. Table 3-1 shows these

dynamics and the augmentation required. Also shown are the bare and augmented short

period natural frequencies (ωsp) and damping ratios (ζsp) for each case except for Case D

which is unstable. The time to double amplitude (T2) is shown instead. The actuator rate

limits used in this project were 15 deg/sec, 30 deg/sec, 45 deg/sec and 60 deg/sec. The

Matlab/SimulinkTM diagram used to determine the bare and augmented aircraft transfer

functions is shown in Figure 3-4. Appendix A contains the fourth order state space

model matrices of the linear decoupled small perturbation longitudinal equations of

motion.

Table 3-1. HAVE PREVENT Case Characteristics (Hanley, 2003)

Case Bare Aircraft Poles ωsp ζsp Kq Kα

Aircraft Poles with Stability Augmentation

A -0.017 ± 0.074j -2.200 ± 2.220j 3.125 0.704 0 0 -0.017 ± 0.074j

-2.200 ± 2.220j

B -0.016 ± 0.079j -1.420 ± 1.860j 2.34 0.61 0.14 0.21 -0.0166 ± 0.0736j

-2.261 ± 2.359j

C -0.009 ± 0.097j -0.860 ± 0.084j 0.86 0.995 0.24 0.51 -0.0168 ± 0.0737j

-2.241 ± 2.517j

D -0.017 ± 0.033j 1.07, -1.67 T2 = 2.31 sec 0.34 0.61 -0.0169 ± 0.0737j

-2.317 ± 2.624j

3-4

Figure 3-4. Longitudinal State Space Diagram (Matlab/SimulinkTM, 2001)

Gap Criterion Calculation for HAVE PREVENT Datasets.

Step 1.

The pitch-to-actuator transfer functions for the bare aircraft dynamics are shown

in Table 3-2. They were calculated using the MatlabTM code shown in Appendix B.

Table 3-2. HAVE PREVENT Pitch-to-Actuator Transfer Functions ( cG )

Case cG

A 2

4 3 2

11.09 14.37 0.52774.402 9.889 0.3562 0.05612

s ss s s s

− − −+ + + +

B 2

4 3 2

11.09 14.37 0.52772.887 5.573 0.1938 0.03557

s ss s s s

− − −+ + + +

C 2

4 3 2

11.09 14.37 0.52771.729 0.7799 0.02955 0.007019

s ss s s s

− − −+ + + +

D 2

4 3 2

11.08 14.37 0.52770.634 1.765 0.05993 0.002462

s ss s s s

− − −+ − − −

Step 2.

Since Case D has unstable short period poles, the Gap Criterion is automatically

set equal to zero for all rate limits.

3-5

Step 3.

Actuator dynamics for this dataset are 2020actuatorG

s=

+.

Step 4.

The Neal-Smith pilot model transfer functions are shown in Table 3-3 along with

the augmented dynamics for which they were computed. They were calculated using the

Interactive Flying Qualities Toolbox for Matlab provided by the Flying Dynamics

Directorate of the Air Force Research Laboratories (Doman and Forringer, 1996).

Table 3-3. HAVE PREVENT Neal-Smith Pilot Models Case pG *actuator augmentedG G

A ( ) ( )

( )0.255 1 0.31659 1

0.125330.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5524.4 97.93 198.1 7.179 1.122

s ss s s s s

− − −+ + + + +

B ( ) ( )

( )0.255 1 0.28336 1

0.126520.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5522.89 94.57 198.8 7.018 1.114

s ss s s s s

− − −+ + + + +

C ( ) ( )

( )0.255 1 0.25433 1

0.126180.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5521.73 89.11 198.6 7.088 1.118

s ss s s s s

− − −+ + + + +

D ( ) ( )

( )0.255 1 0.23182 1

0.126590.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5520.63 86.93 198.7 7.12 1.12

s ss s s s s

− − −+ + + + +

Step 5 and 6.

Figure 3-5 shows the Nichols charts for Cases A, B and C.

Step 7.

Cases A and B are of Type I. Case C is Type III.

3-6

Case A

Case B Case C

Figure 3-5. HAVE PREVENT Nichols Charts for Cases A, B and C

Type I, Step 1-1 through 1-2.

The minimum amount of effective increase in gain for Cases A and B, along with

the determined values for K*, ω and Amax, are shown in Table 3-4:

Table 3-4. HAVE PREVENT Components for Cases A and B

Case ( )pK dB∆ K* ω(rad/s) Amax (deg) Case A 8.431 0.829 4.51 30 Case B 3.159 0.726 2.80 30

The final values for the Gap Criterion and amplitude for each case and rate limit

are shown in Table 3-5:

3-7

Table 3-5. Gap Criteria Values for HAVE PREVENT Cases A and B

Case Maximum Actuator Deflection Rate (VL, deg/sec) A (deg) Gap

Criterion 15 6.302 0.555 30 12.604 1.109 45 18.906 1.664 A

60 25.208 2.218 15 11.591 0.556 30 23.182 1.112 45 34.773 1.667 B

60 46.364 2.223

Type III, Step 3-1 through 3-4.

The determined values for K*, ω, Amax and Gap Criterion for each rate limit

applied to Case C are shown in Table 3-6:

Table 3-6. Gap Criteria for HAVE PREVENT Case C

Case K* ω(rad/s) Amax (deg) Maximum Actuator Deflection Rate (VL, deg/sec) A(deg) Gap

Criterion15 9.002 0.300 30 18.004 0.600 45 27.006 0.900 C 0.999 2.62 30

60 36.008 1.200

HAVE PREVENT Gap Criterion Summary.

Table 3-7 summarizes the Gap Criteria for Project HAVE PREVENT. The

dataset produced a range of Gap Criterion from 0.0 to 2.233. The range was well

distributed for attempting to correlate Gap Criterion and PIOR.

3-8

Table 3-7. HAVE PREVENT Gap Criteria Summary

Case Maximum Actuator Deflection Rate (VL, deg/sec)

Gap Criterion

15 0.555 30 1.109 45 1.664 A

60 2.218 15 0.556 30 1.112 45 1.667 B

60 2.223 15 0.300 30 0.600 45 0.900 C

60 1.200 15 0.0 30 0.0 45 0.0 D

60 0.0

HAVE PREVENT Gap Criterion Correlation.

The PIOR and Gap Criterion data were paired and the results plotted in Figures 3-

6 through 3-8. There were cases in which the same bare aircraft dynamics and rate limts

and hence the same Gap Criterion value were tested multiple times by the same pilot or

different pilots. In these cases, the same PIOR was often found. This resulted in multiple

pairs of datapoints with the same Gap Criterion and PIOR vaules. In an effort to

represent this data density, all plots will use a “bubble chart” style in which bubble size

indicates multiple datapoints with the same pair values.

3-9

y = -1.5313x + 4.8615R2 = 0.3427|R| = 0.5854

-0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4

Gap Criterion

PIO

Ten

denc

y R

atin

g Single Data Point

Two Data Points

Three Data Points

Four Data Points

Linear Fit

1

2

3

4

5

6

Data Basis: 15K ft PA, 300 KIASTest A/C: LAMARS SimulationAcquisition System: Hand HeldConfiguration: CruiseTest Dates: 25-26 September 2003

Figure 3-6. HAVE PREVENT Phase 2 Sum-of-sines Task LAMARS Data

y = -1.0832x + 4.1378R2 = 0.3555|R| = 0.5962

-0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Linear Fit


1

2

3

4

5

6

Figure 3-7. HAVE PREVENT Phase 3 Discrete HUD Pitch Tracking Task LAMARS

Data

3-10

y = -0.6356x + 3.8593R2 = 0.1549|R| = 0.3936

-0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4

Gap Criterion

PIO

Ten

denc

y R

atin

g Single Data Point

Two Data Points

Linear Fit


1

2

3

4

5

6

Figure 3-8. HAVE PREVENT Phase 3 Target Tracking Task LAMARS Data

With just a glance, it is difficult to see any readily apparent trend in the data

scatter. Therefore, the probability of a linear correlation existing for the data was

investigated as described in Appendix C (Wheeler and Ganji, 1996:145-147). The linear

fit depicted on the plots displays this correlation.

Confidence level was used to determine if a relationship existed between PIOR

and corresponding Gap Criterion values. Common engineering practices suggest a

confidence level of 95% represents definitive “real” correlation (Wheeler and Ganji,

1996:147). A table listing the linear correlation coefficient, minimum correlation

coefficient for 95% confidence and actual confidence level for each dataset is shown

below. It can be seen that 2 out of 3 datasets easily met the minimum confidence level.

The Phase 3 Target Tracking dataset has fewer datapoints than the other two and hence

will be more sensitive to outliers.

3-11

Table 3-8. HAVE PREVENT Correlation Confidence Levels

Dataset Sample Size

95 % Confidence Minimum

Correlation

Actual Linear Correlation

Actual Confidence Level

Phase 2 Sum-of-sines 30 0.361 0.585 99.93%

Phase 3 Discrete Pitch Tracking 19 0.456 0.596 99.29%

Phase 3 Target Tracking 12 0.576 0.394 79.49%

HAVE PREVENT Best Curve Fit.

In the previous section, correlation for a linear fit was investigated. This is not

necessarily the best curve fit. Multiple curve fits were attempted for each dataset

including exponential, natural logarithmic, polynomial and power series interpolations.

A least squares method was used to identify the correlation coefficient of each fit using

commercial software (Microsoft Excel, 2002). The following table lists these correlation

values.

Table 3-9. HAVE PREVENT Curve Fit Correlation Values

Dataset Linear Exponential Logarithmic (Ln (GC+1))

2nd Order Polynomial

Power Series (GC+1)

Phase 2 Sum-of-sines 0.585 0.561 0.552 0.593 0.525

Phase 3 Discrete Pitch Tracking 0.596 0.507 0.672 0.786 0.577

Phase 3 Target Tracking 0.394 0.426 0.335 0.468 0.357

At first glance, the second order polynomial curve fit gives the best mathematical

results. But on further inspection, this curve fit does not match well at higher Gap

Criterion values. The logarithmic (Ln(GC+1)) seems to be the best case for the

3-12

combination of high correlation and endpoint matching. Since Gap Criterion (GC)

values of zero do not work well with natural logarithmic curve fits, a factor of one was

added to these values and the correlation found. Figures 3-6 through 3-8 show the data

with this logarithmic curve fit.

-0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Logarithmic Fit

1

2

3

4

5

6


y = -2.7585 Ln(GC+1) + 5.0677R = 0.3048|R| = 0.552

2

Figure 3-9. HAVE PREVENT Phase 2 Sum-of-sines Task LAMARS Data with

Logarithmic Curve Fit

3-13

-0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4

Gap Criterion

PIO

Ten

denc

y R

atin

g Single Data Point

Two Data Points

Logarithmic Fit


1

2

3

4

5

6

y = -2.42 Ln(GC+1) + 4.6113R = 0.4522|R| = 0.6725

2

Figure 3-10. HAVE PREVENT Phase 3 Discrete HUD Pitch Tracking Task LAMARS

Data with Logarithmic Curve Fit

-0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Logarithmic Fit


1

2

3

4

5

6

y = -1.0052 Ln(GC+1) + 3.8443R = 0.1124|R| = 0.3353

2

Figure 3-11. HAVE PREVENT Phase 3 Target Tracking Task LAMARS Data with


3-14

HAVE OLOP Analysis

HAVE OLOP was a flight test study attempting to validate the Open Loop Onset

Point (OLOP) criteria for PIO prediction. The study was conducted in the VISTA NF-

16D as part of a Test Management Project at the USAF Test Pilot School and a thesis

sponsored by the Air Force Institute of Technology (Gilbreath and others, 2000;

Gilbreath, 2001). Just as in Project HAVE PREVENT, PIOR were gathered in two

distinct phases with specific piloting tasks. In fact, with the exception of the Phase 3

Target Tracking Task which was not performed, the tasks were identical to those of

HAVE PREVENT: Phase 2 Sum-of-sines Tracking Task and Phase 3 Discrete HUD

Pitch Tracking Task. The PIOR scale used was the same one previously described in

Chapter I.

This study also involved four test cases labeled A through D though each was

distinctly different from the identically named HAVE PREVENT cases. Each test case

had different bare aircraft dynamics but all were augmented with angle-of-attack and

pitch-rate feedback to produce almost identical closed loop dynamics. Table 3-10 shows

these dynamics and the augmentation required. Also shown are the bare and augmented

short period natural frequencies (ωsp) and damping ratios (ζsp) for each case except for

Case D which is unstable and diverges. Instead, the time to double amplitude (T2) is

shown. The actuator rate limits used in this project were 10 deg/sec, 20 deg/sec, 30

deg/sec, 40 deg/sec, 50 deg/sec and 60 deg/sec. The Matlab/SimulinkTM diagram used to

determine the bare and augmented aircraft transfer functions is shown in Figure 3-4.

Appendix A contains the fourth order state space model matrices of the linear decoupled

small perturbation longitudinal equations of motion.

3-15

Table 3-10. HAVE OLOP Case Characteristics (Gilbreath, 2001)



A -.017±.074j -.510±3.36j 3.40 0.15 0.023 0.412 -.017±.060j

-3.51±-3.58j

B -.017±.074j -.608±.488j 0.78 0.78 0.444 0.984 -.017±.072j

-3.50±3.58j

C -.017±.074j -1.29±1.72j 2.15 0.60 0.897 0.347 -.017±.074j

-3.50±3.57j

D -.017±.074j 1.284, -2.13 T2 = 0.6sec 1.218 0.487 -.017 ±.074j

-3.55±3.62j

Gap Criterion Calculation for HAVE OLOP Datasets.

Step 1.

The bare aircraft dynamics are shown in Table 3-11:

Table 3-11. HAVE OLOP Pitch-to-Actuator Transfer Functions ( cG )

Case cG

A 2

4 3 2

11.08 14.37 0.52771.054 11.60 0.3989 0.06664

s ss s s s

− − −+ + + +

B 2

4 3 2

11.09 14.37 0.52771.251 0.6555 0.0277 0.003507

s ss s s s

− − −+ + + +

C 2

4 3 2

11.08 14.37 0.52772.614 4.716 0.172 0.02665

s ss s s s

− − −+ + + +

D 2

4 3 2

11.08 14.37 0.52770.88 2.7 0.08811 0.01577

s ss s s S

− − −+ − − −

Step 2.

Since Case D has unstable short period poles, the Gap Criterion is automatically

set equal to zero for all rate limits.

3-16

Step 3.

Actuator dynamics for this dataset are 2020actuatorG

s=

+.

Step 4.


the augmented dynamics for which they were computed: They were calculated using the

Interactive Flying Qualities Toolbox for Matlab (Doman and Forringer, 1996).

Table 3-12. HAVE OLOP Neal-Smith Pilot Models Case pG *actuator augmentedG G

A ( ) ( )

( )0.255 1 0.07543 1

0.231080.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5521.05 124.1 356.0 12.57 1.377

s ss s s s s

− − −+ + + + +

B ( ) ( )

( )0.255 1 0.074331 1

0.233980.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5521.25 125.2 360.7 12.88 1.956

s ss s s s s

− − −+ + + + +

C ( ) ( )

( )0.255 1 0.068821 1

0.252670.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5522.61 134.9 394.8 14.1 2.253

s ss s s s s

− − −+ + + + +

D ( ) ( )

( )0.255 1 0.072622 1

0.232130.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5520.88 124.1 357.9 12.81 2.018

s ss s s s s

− − −+ + + + +

Step 5 and 6.

Figure 3-12 shows the Nichols charts for Cases A, B and C:

Step 7.

Cases A and C are of Type II. Case B is Type IV.

3-17

Case A

Case B Case C

Figure 3-12. HAVE OLOP Nichols Charts for Cases A, B and C

Type II, Step 2-1 through 2-2.

The minimum amount of effective decrease in gain for Cases A and C, along with


Table 3-13. HAVE OLOP Components for Cases A and C Case ( )pK dB∆ K* ω(rad/s) Amax (deg)

Case A -10.290 0.943 3.50 30 Case C -5.455 0.801 2.05 30

3-18

The final values for the Gap Criteria and amplitudes for Cases A and C and their

respective rate limits are shown in Table 3-14:

Table 3-14. Gap Criteria Values for HAVE OLOP Case A and C


Criterion 10 4.76 0.0485 20 9.52 0.0970 30 14.28 0.1456 40 19.04 0.1941 50 23.80 0.2426

A

60 28.55 0.2911 10 9.57 0.1702 20 19.13 0.3403 30 28.70 0.5105 40 38.26 0.6806 50 47.83 0.8508

C

60 57.40 1.0210

Type IV, Step 4-1.

HAVE OLOP Case B is a Type IV and has no Gap Criterion solution.

3-19

HAVE OLOP Gap Criterion Summary.

Table 3-15 summarizes the Gap Criteria for Project HAVE OLOP:

Table 3-15. HAVE OLOP Gap Criteria Summary


Gap Criterion

10 0.0485 20 0.0970 30 0.1456 40 0.1941 50 0.2426

A

60 0.2911 10 0.1702 20 0.3403 30 0.5105 40 0.6806 50 0.8508

C

60 1.0210 10 0.0 20 0.0 30 0.0 40 0.0 50 0.0

D

60 0.0

HAVE OLOP Gap Criterion Correlation.

The PIOR and Gap Criterion data were paired and the results plotted in Figures 3-

13 and 3-14:

3-20

y = -1.4253x + 4.8321R2 = 0.2276|R| = 0.4771

-0.25 0 0.25 0.5 0.75 1 1.25

Gap Criterion

PIO

Ten

denc

y R

atin

gSingle Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Sixteen Data Points

Linear Fit

Data Basis: 15K ft PA, 300 KIASTest A/C: NF-16D - # 86-00048Acquisition System: Hand HeldConfiguration: Cruise / VSS EngagedTest Dates: 3-16 October 2003

1

2

3

4

5

6

Figure 3-13. HAVE OLOP Phase 2 Sum-of-sines Task VISTA Data

y = -1.5299x + 4.0208R2 = 0.2119|R| = 0.4603

-0.25 0 0.25 0.5 0.75 1 1.25

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Twelve Data Points

Linear Fit


1

2

3

4

5

6

Figure 3-14. HAVE OLOP Phase 3 Discrete HUD Pitch Tracking Task VISTA Data

3-21

As in the HAVE PREVENT data analysis, the probability of a linear correlation

existing was investigated as described in Appendix C. The linear fit depicted on the plots

displays this correlation.

Confidence level was used to determine if a relationship existed between PIOR

and corresponding Gap Criterion values with a confidence level of 95% representing

definitive “real” correlation (Wheeler and Ganji, 1996:147). A table listing the linear

correlation coefficient, minimum correlation coefficient and confidence level for each

dataset is shown below. It can be seen that both datasets easily met the “real” confidence

level.

Table 3-16. HAVE OLOP Correlation Confidence Levels

Dataset Sample Size


Correlation





HAVE OLOP Best Curve Fit.


necessarily the best curve fit. As in the HAVE PREVENT data analysis, multiple curve

fits were attempted for each dataset including exponential, natural logarithmic,

polynomial and power series interpolations. A least squares method was used to identify

the correlation coefficient of each fit as calculated for the HAVE PREVENT data. The

following table lists these correlation values:

3-22

Table 3-17. HAVE OLOP Curve Fit Correlation Values

Dataset Linear Exponential Logarithmic (Ln (GC+1))

2nd Order Polynomial

Power Series (GC+1)



Again, the second order polynomial curve fit gives the best mathematical results.

But on further inspection, this curve fit still does not match well at higher Gap Criterion

values. The natural logarithmic (Ln(GC+1)) seems to be the best case for the

combination of high correlation and endpoint matching. For the same reasons as in the

HAVE PREVENT curve fitting, a factor of one was added to the Gap Criterion values

and the correlation found. Figures 3-15 and 3-16 show the data with this logarithmic

curve fit.

3-23

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Gap Criterion

PIO

Ten

denc

y R

atin

gSingle Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Sixteen Data Points

Logarithmic Fit


1

2

3

4

5

6

y = -2.0909 Ln(GC+1) + 4.8918R = 0.2452|R| = 0.4952

2

Figure 3-15. HAVE OLOP Phase 2 Sum-of-sines Task VISTA Data with Logarithmic

Fit

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Twelve Data Points

Logarithmic Fit


1

2

3

4

5

6

y = -2.2936 Ln(GC +1) + 4.0989R = 0.2433|R| = 0.4933

2

Figure 3-16. HAVE OLOP Phase 3 Discrete HUD Pitch Tracking Task VISTA Data

with Logarithmic Fit

3-24

HAVE PREVENT and HAVE OLOP Summary

Each dataset from both studies shows correlation. All Phase 2 Sum-of-sines and

Phase 3 Discrete HUD Pitch Tracking Task datasets had linear correlation confidence

levels better than 99%. The HAVE PREVENT Phase 3 Target Tracking Task dataset had

the worst correlation confidence level, never exceeding 87.5% for any of the curve fits.

4-1

IV Analysis of Project MAX GAP (LAMARS) Data

Project MAX GAP sought to augment the HAVE PREVENT and HAVE OLOP

datasets. The same Phase 2 Sum-of-sines Tracking Task, Phase 3 Discrete HUD Pitch

Tracking Task and Phase 3 Target Tracking Task described in Chapter III were used in

the LAMARS simulator. The PIOR scale used was the same one previously described.

MAX GAP (LAMARS) Analysis

The study involved one previous test case, Case B from HAVE PREVENT, and

three new test cases labeled N, W and Y. A series of bare aircraft dynamics with varying

short-period characteristics were chosen (26 in all). The Gap Criterion was applied to

each of these. Cases B, N, W and Y were chosen for their Gap Criterion distribution.

Each test case was augmented with angle-of-attack and pitch-rate feedback which

produced almost identical closed-loop dynamics. Table 4-1 shows these dynamics and

the augmentation required. Also shown are the bare and augmented short period natural

frequencies (ωsp) and damping ratios (ζsp) for each case. The actuator rate limits used in

this project were 15 deg/sec, 30 deg/sec and 60 deg/sec. See Appendix A for the fourth

order state space model matrices.

Table 4-1. MAX GAP (LAMARS) Case Characteristics (Witte and others, 2003)



B -1.43±1.85j -.017±.074j 2.34 0.61 0.156 0.123 -2.17±-2.22j

-.017±.070j

N -.939±2.99j -.017±.074j 3.13 0.30 -0.335 0.177 -2.18±2.22j

-.018±.050j

W -4.24±2.05j -.017±.074j 4.71 0.90 -0.501 -0.345 -2.18±2.24j

-.017 ±.081j

Y -2.09±1.01j -.017±.074j 2.32 0.90 0.346 0.0334 -2.19±2.24j

-.017±.080j

4-2

Gap Criterion Calculation for MAX GAP (LAMARS) Datasets.

Step 1.


in Table 4-2 and were computed using the MatlabTM code in Appendix B.

Table 4-2. MAX GAP (LAMARS) Pitch-to-Actuator Transfer Functions ( cG )

Case cG

B 2

4 3 2

11.08 14.37 0.52772.889 5.578 0.2026 0.03157

s ss s s s

− − −+ + + +

N 2

4 3 2

11.09 14.37 0.52771.912 9.867 0.3439 0.05648

s ss s s s

− − −+ + + +

W 2

4 3 2

11.09 14.37 0.52778.512 22.48 0.8031 0.1279

s ss s s s

− − −+ + + +

Y 2

4 3 2

11.08 14.37 0.52774.21 5.53 0.2071 0.03103

s ss s s s

− − −+ − − −

Step 2.

Does not apply; all four cases have stable poles for their bare aircraft dynamics.

Step 3.

Actuator dynamics for this dataset were 2020actuatorG

s=

+.

Step 4.


the augmented dynamics for which they were computed. They were calculated using the

Interactive Flying Qualities Toolbox for Matlab provided by the Flying Dynamics

Directorate of the Air Force Research Laboratories (Doman and Forringer, 1996).

4-3

Table 4-3. MAX GAP (LAMARS) Neal-Smith Pilot Models Case pG *actuator augmentedG G

B ( ) ( )

( )0.255 1 0.32699 1

0.114830.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5522.89 90.88 182.1 6.597 0.9303

s ss s s s s

− − −+ + + + +

N ( ) ( )

( )0.255 1 0.32699 1

0.109190.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5521.91 86.89 173.2 6.186 0.4845

s ss s s s s

− − −+ + + + +

W ( ) ( )

( )0.255 1 0.30311 1

0.152540.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5528.51 116.3 239.7 8.681 1.596

s ss s s s s

− − −+ + + + +

Y ( ) ( )

( )0.255 1 0.31318 1

0.125280.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5524.21 97.5 197.9 7.218 1.286

s ss s s s s

− − −+ + + + +

Step 5 and 6.

Figure 4-1 shows the Nichols charts for Cases B, N, W and Y:

Case B Case N

Case W Case Y

Figure 4-1. MAX GAP (LAMARS) Nichols Charts for Cases B, N, W and Y

4-4

Step 7.

Cases B, N, W and Y are all of Type I.


The minimum amount of effective increase in gain for Cases B, N, W and Y,

along with the determined values for K*, ω and Amax, are shown in Table 4-4. The final

values for the Gap Criterion and amplitude for each case and rate limit are shown in

Table 4-5

Table 4-4. MAX GAP (LAMARS) Components for Cases B, N, W and Y

Case ( )pK dB∆ K* ω(rad/s) Amax (deg) Case B 4.065 0.719 2.93 30 Case N 1.661 0.806 3.70 30 Case W 12.450 0.955 7.07 30 Case Y 7.230 0.664 2.95 30

Table 4-5. Gap Criteria Values for MAX GAP (LAMARS) Cases B, N, W and Y


Criterion 15 11.18 0.5953 30 22.37 1.1907 B 60 44.72 2.3814 15 7.90 0.3189 30 15.80 0.6377 N 60 31.60 1.2754 15 3.49 0.4880 30 6.98 0.9760 W 60 13.95 1.9520 15 12.03 0.9276 30 24.06 1.8552 Y 60 48.12 3.7104

4-5

MAX GAP (LAMARS) Gap Criterion Summary.

Table 4-6 summarizes the LAMARS Gap Criteria for Project MAX GAP:

Table 4-6. MAX GAP (LAMARS) Gap Criteria Summary


Gap Criterion

15 0.5953 30 1.1907 B 60 2.3814 15 0.3189 30 0.6377 N 60 1.2754 15 0.4880 30 0.9760 W 60 1.9520 15 0.9276 30 1.8552 Y 60 3.7104

MAX GAP (LAMARS) Gap Criterion Correlation.

The MAX GAP test team recognized that some outlier datapoints were included

in the LAMARS datasets. In these cases, poor pilot recognition of PIO cues such as

aircraft lag response had resulted in improper PIOR assessment. Post-test analysis

revealed these datapoints which were then omitted from the “reduced dataset” findings

(Witte and others, 2003:8). These reduced LAMARS datasets were used for this study.

The remaining PIOR and Gap Criterion data were paired and the results plotted in

Figures 4-2 through 4-4:

4-6

y = -1.0794x + 4.915R2 = 0.3376|R| = 0.581

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Eight Data Points

Linear Fit

Data Basis: 15K ft PA, 300 KIASTest A/C: LAMARS SimulationAcquisition System: Hand HeldConfiguration: CruiseTest Dates: 22 August 2003

1

2

3

4

5

6

Figure 4-2. MAX GAP (LAMARS) Phase 2 Sum-of-sines Task Data

y = -0.7468x + 4.3429R2 = 0.2509|R| = 0.5009

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Points

Two Data Points

Three Data Points

Four Data Points

Linear Fit


1

2

3

4

5

6

Figure 4-3. MAX GAP (LAMARS) Phase 3 Discrete HUD Pitch Tracking Task Data

4-7

y = -0.6934x + 3.8932R2 = 0.2197|R| = 0.4687

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Linear Fit

1

2

3

4

5

6


Figure 4-4. MAX GAP (LAMARS) Phase 3 Target Tracking Task Data

As in the HAVE PREVENT and HAVE OLOP data analyses, the probability of a

linear correlation existing for the data was investigated and is described in Appendix C.

The linear fit depicted on the plots displays this correlation.

Again, confidence level was used to determine if a relationship existed between

PIOR and corresponding Gap Criterion values with a confidence level of 95%

representing definitive “real” correlation (Wheeler and Ganji, 1996:147). A table listing

the linear correlation coefficient, minimum correlation coefficient and confidence level

for each the datasets of the LAMARS portion of Project MAX GAP is shown below. It

can be seen that all three datasets exceeded the necessary confidence level.

4-8

Table 4-7. MAX GAP (LAMARS) Correlation Confidence Levels

Dataset Sample Size


Correlation





Phase 3 Target Tracking 44 0.297 0.469 99.87%

MAX GAP (LAMARS) Best Curve Fit.


necessarily the best curve fit. Multiple curve fits were attempted for each dataset

including exponential, natural logarithmic, polynomial and power series interpolations

just as in the HAVE PREVENT and HAVE OLOP data analyses. A least squares method

was used to identify the correlation coefficient of each fit. Table 4-9 lists these

correlation values:

Table 4-8. MAX GAP (LAMARS) Curve Fit Correlation Values

Dataset Linear Exponential Logarithmic 2nd Order Polynomial Power Series



Phase 3 Target Tracking 0.469 0.485 0.482 0.482 0.489

Again, the second order polynomial curve fit gives the best mathematical results.

But this curve fit still does not match well at higher Gap Criterion values. Just as in the

4-9

HAVE PREVENT and HAVE OLOP analyses, the natural logarithmic curve fit seems to

be the best case for the combination of high correlation and endpoint matching. There

were no Gap Criterion values of zero in this analysis and therefore simple natural

logarithmic curve fitting was accomplished. Figures 4-5 and 4-6 show the data with this

logarithmic curve fit.

y = -1.3432Ln(x) + 3.5931R2 = 0.3285|R| = 0.5731

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Eight Data Points

Logarithmic Fit


1

2

3

4

5

6

Figure 4-5. MAX GAP (LAMARS) Phase 2 Sum-of-sines Task Data with Logarithmic

Curve Fit

4-10

y = -0.9276Ln(x) + 3.4468R2 = 0.2568|R| = 0.5067

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Points

Two Data Points

Three Data Points

Four Data Points

Linear Fit


1

2

3

4

5

6

Figure 4-6. MAX GAP (LAMARS ) Phase 3 Discrete HUD Pitch Tracking Task Data

with Logarithmic Fit

y = -0.9677Ln(x) + 3.0339R2 = 0.2319|R| = 0.4816

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Linear Fit

1

2

3

4

5

6


Figure 4-7. MAX GAP (LAMARS) Phase 3 Target Tracking Task Data with

Logarithmic Fit

4-11

MAX GAP (LAMARS) Summary

Each dataset shows correlation and each linear correlation confidence level is

better than 99%. The HAVE PREVENT and HAVE OLOP data analyses bore out the

fact that correlation exists between the Gap Criterion and PIOR and the LAMARS

portion of Project MAX GAP reinforces this assertion.

5-1

V. Analysis of Project MAX GAP (VISTA) Data

More Gap Criterion and PIO tendency rating was acquired in the flight test

portion of Project MAX GAP. The VISTA NF-16D was used to accomplish this. Again,

the same Phase 2 Sum-of-sines Tracking Task, Phase 3 Discrete HUD Pitch Tracking

Task and Phase 3 Target Tracking Task described in Chapter III were used. The PIOR

scale used was the MIL-HDBK-1797 one previously described in Chapter I.

MAX GAP (VISTA) Analysis

The same four sets of bare aircraft dynamics and the same three actuator rate

limits used in the MAX GAP LAMARS test were requested for the VISTA aircraft. The

actuator rate limits were easily programmed, but, unfortunately, obtaining exact matches

in the VISTA NF-16D is as much an art as a science. Compounding the problem, only

one calibration sortie was accomplished due to project time constraints. The bare aircraft

dynamics for Cases B and N were suitably close to the specified values, but the

augmented dynamics for Case B were significantly different. The bare aircraft dynamic

damping ratio for Case N was almost perfect, but the requested natural frequency was

less by 1 rad/sec. Case Y proved to be an overdamped though stable system. The short

period characteristics of these cases were used for the actual Gap Criterion analysis and

are shown in Table 5-1.

As described in the test report for Project MAX GAP, poor task setup and

execution were experienced in the Phase 3 Target Tracking Task. The T-38A target

aircraft was allowed to develop too much angle away from the VISTA prior to an

engagement being conducted. As a result of this initial offset and less than expected turn

performance in the VISTA, the gross acquisition task turned into a low gain task, rather

5-2

than the high gain task of a Category A maneuver (Witte and others, 2003:10). For these

reasons, this dataset was ignored in this study. Additionally, the MAX GAP test team

found instances where the evaluation pilots missed critical PIO cues such as aircraft lag

response and assessed improper PIOR. These data points were omitted in the “reduced

dataset” results of Project MAX GAP. These reduced datasets were used for this study.

Table 5-1. MAX GAP (VISTA) Case Characteristics (Witte and others, 2003)

Case Bare Aircraft Poles ωsp ζsp Kq Kα Aircraft Poles with

Stability Augmentation

B -1.42±1.85j -.017±.074j 2.33 0.61 -0.0212 0.0877 -1.98±-1.69j

-.017±.065j

N -.493±2.86j -.017±.074j 2.90 0.17 -0.408 0.254 -2.29±1.95j

-.018±.031j

W -3.26±1.77j -.017±.074j 3.70 0.88 -0.0946 -0.185 -2.10±2.14j

-.017 ±.082j

Y -3.02, -0.96 -.017±.074j 1.70 1.17 0.485 0.0400 -2.09±2.14j

-.017±.083j

Gap Criterion Calculation for MAX GAP (VISTA) Datasets.

Step 1.


in Table 5-2:

Table 5-2. MAX GAP (VISTA) Pitch-to-Actuator Transfer Functions ( cG )

Case cG

B 2

4 3 2

11.08 14.37 0.52772.877 5.531 0.201 0.0313

s ss s s s

− − −+ + + +

N 2

4 3 2

11.08 14.37 0.52771.02 8.449 0.2916 0.04848

s ss s s s

− − −+ + + +

W 2

4 3 2

11.09 14.37 0.52776.546 13.92 0.503 0.07892

s ss s s s

− − −+ + + +

Y 2

4 3 2

11.08 14.37 0.52774.012 3.031 0.1212 0.01666

s ss s s s

− − −+ + + +

5-3

Step 2.

Does not apply; all four cases have stable poles for their bare aircraft dynamics.

Step 3.

Actuator dynamics for this dataset were 2020actuatorG

s=

+.

Step 4.


the augmented dynamics for which they were computed (Doman and Forringer, 1996):

Table 5-3. MAX GAP (VISTA) Neal-Smith Pilot Models Case pG *actuator augmentedG G

B ( ) ( )

( )0.255 1 0.53135 1

0.0789970.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5522.88 82.48 131.3 4.811 0.5853

s ss s s s s

− − −+ + + + +

N ( ) ( )

( )0.255 1 0.42704 1

0.095310.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5521.02 84.65 151.1 5.496 0.5768

s ss s s s s

− − −+ + + + +

W ( ) ( )

( )0.255 1 0.34451 1

0.128460.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5526.55 103.6 204.4 7.448 1.397

s ss s s s s

− − −+ + + + +

Y ( ) ( )

( )0.255 1 0.35487 1

0.112930.0001 1

ss se

s s−+ +

−+

2

5 4 3 2

221.7 287.5 10.5524.01 92.64 180.6 6.626 1.263

s ss s s s s

− − −+ + + + +

Step 5 and 6.

Figure 5-1 shows the Nichols charts for Cases B, N, W and Y:

Step 7.

Cases B and W are of Type I, Case N is Type II and Case Y is Type III.

5-4

Case B Case N

Case W Case Y

Figure 5-1. MAX GAP (VISTA) Nichols Charts for Cases B, N, W and Y


The minimum amount of effective increase in gain for Cases B and W along with


Table 5-4. MAX GAP (VISTA) Components for Cases B and W Case ( )pK dB∆ K* ω(rad/s) Amax (deg)

Case B 6.457 0.736 3.57 30 Case W 11.257 0.922 6.12 30

5-5

The final values for the Gap Criterion and amplitude for these two cases and rate

limits are shown in Table 5-5:

Table 5-5. Gap Criteria Values for MAX GAP (VISTA) Cases B and W


Criterion 15 8.97 0.6287 30 17.93 1.2573 B 60 35.87 2.5146 15 4.18 0.5129 30 8.35 1.0257 W 60 16.70 2.0515

Type II, Step 2-1 through 2-2.

The minimum amount of effective decrease in gain for Case N, along with the

determined values for K*, ω and Amax, are shown in Table 5-6:

Table 5-6. MAX GAP (VISTA) Components for Case N Case ( )pK dB∆ K* ω(rad/s) Amax (deg)

Case N -3.998 0.784 3.26 30

The final values for the Gap Criteria and amplitudes for Case N and its respective

rate limits are shown in Table 5-7:

Table 5-7. Gap Criteria Values for MAX GAP (VISTA) Case N


Criterion 15 9.22 0.1939 30 18.44 0.3879 N 60 36.88 0.7758

5-6

Type III, Step 3-1 through 3-4.

The determined values for K*, ω, Amax and Gap Criterion for each rate limit

applied to Case Y are shown in Table 5-8:

Table 5-8. Gap Criteria for MAX GAP (VISTA) Case Y

Case K* ω(rad/s) Amax (deg) A(deg)

Maximum Actuator Deflection Rate (VL,

deg/sec)

Gap Criterion

40.40 15 1.3467 80.80 30 2.6934 Y 0.540 1.08 30 161.60 60 5.3868

MAX GAP (VISTA) Gap Criterion Summary.

Table 5-9 summarizes the Gap Criteria for the flight test portion of Project MAX

GAP.

Table 5-9. MAX GAP (VISTA) Gap Criteria Summary


Gap Criterion

15 0.6287 30 1.2573 B 60 2.5146 15 0.1939 30 0.3879 N 60 0.7758 15 0.5129 30 1.0257 W 60 2.0515 15 1.3467 30 2.6934 Y 60 5.3868

MAX GAP (VISTA) Gap Criterion Correlation.

The PIOR and Gap Criterion data were paired and plotted in Figures 5-2 and 5-3:

5-7

y = -0.4536x + 3.975R2 = 0.1601|R| = 0.4001

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Linear Fit


1

2

3

4

5

6

Figure 5-2. MAX GAP (VISTA) Phase 2 Sum-of-sines Task Data

y = -0.2334x + 3.624R2 = 0.1094|R| = 0.3308

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Linear Fit


1

2

3

4

5

6

Figure 5-3. MAX GAP (VISTA) Phase 3 Discrete HUD Pitch Tracking Task Data

5-8

As in the previous analyses, the probability of a linear correlation existing for the

data was investigated as described in Appendix C. The linear fit depicted on the plots

displays this correlation.

Again, confidence level was used to determine if a relationship existed between

PIOR and corresponding Gap Criterion values. A 95% confidence level was maintained

as the threshold for justifiably acceptable correlation (Wheeler and Ganji, 1996:147). A

table listing the linear correlation coefficient, minimum correlation coefficient and

confidence level for both datasets of the flight test portion of Project MAX GAP is shown

below.

Table 5-10. MAX GAP (VISTA) Correlation Confidence Levels

Dataset Sample Size


Correlation





MAX GAP (VISTA) Best Curve Fit.

Multiple curve fits were attempted for these datasets as well. The same

exponential, natural logarithmic, polynomial and power series interpolations were tried.

The correlation coefficient of each fit was calculated and the results are listed below.

5-9

Table 5-11. MAX GAP (VISTA) Curve Fit Correlation Values

Dataset Linear Exponential Logarithmic 2nd Order Polynomial Power Series



Yet again the second order polynomial curve fit gives the highest correlation

values. But this curve fit still does not match well at higher Gap Criterion values. Just as

in the preceding analyses, the natural logarithmic curve fit does a better job of endpoint

matching while maintaining a relatively high correlation factor. Figures 5-4 and 5-5

show the data with this logarithmic curve fit.

y = -0.7408Ln(x) + 3.3278R2 = 0.1711|R| = 0.414

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Logarithmicr Fit


1

2

3

4

5

6

Figure 5-4. MAX GAP (VISTA) Phase 2 Sum-of-sines Task Data with Logarithmic

Curve Fit

5-10

y = -0.4167Ln(x) + 3.2836R2 = 0.1271|R| = 0.357

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Logarithmic Fit


1

2

3

4

5

6

Figure 5-5. MAX GAP (VISTA) Phase 3 Discrete HUD Pitch Tracking Task Data with


MAX GAP (VISTA) Summary

Just is in the MAX GAP (LAMARS) analyses, each dataset shows correlation and

each linear correlation confidence level is better than 95%. The HAVE PREVENT,

HAVE OLOP, MAX GAP (LAMARS) and MAX GAP (VISTA) individually

demonstrate the correlation between Gap Criterion and PIO tendency rating. To

determine if the Gap Criterion can be broadly applied, a correlation within combined

datasets must be accomplished.

In Chapter VI, the datasets are combined and the data are analyzed for correlation

between the PIOR and the Gap Criterion.

6-1

VI. Analysis of Combined Gap Criterion Data

In the previous sections, correlation for individual datasets has been assessed and

best fit curves and data correlation have been established. But, the Gap Criterion is

supposed to be applied universally. The effects of combining datasets will now be

examined. Datasets will be combined for Handling Qualities Phase, task and source.

Finally, simulator and flight test results will be combined for each Handling Qualities

phase and task. Combining Handling Qualities phases was not considered due to

differences in pilot control techniques, specifically HQDT versus operational flying

styles (Brown and others, 2002:21-18 to 21-19).

Figures 6-1 through 6-3 have all datasets obtained from LAMARS plotted

together and Figure6-4 and 6-5 show all VISTA data.

LAMARS Combined Datasets

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Eight Data Points

Logarithmic Fit

HAVE PREVENT Data MAX GAP Data1

2

3

4

5

6y = -2.5964 Ln(GC+1 ) + 5.351

R = 0.3062|R| = 0.5534

Figure 6-1. LAMARS Combined Phase2 Sum-of-sines Dataset

2

6-2

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Points

Two Data Points

Three Data Points

Four Data Points

Logartihmic Fit

1

2

3

4

5

6

HAVE PREVENT Data MAX GAP Data

y = -1.9335 Ln(GC+1 ) + 4.6771R = 0.2863|R| = 0.5351

2

Figure 6-2. LAMARS Combined Phase 3 Discrete (HUD) Pitch-Tracking Dataset

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Logarithmic Fit

1

2

3

4

5

6 y = -1.5918 Ln(GC+1 ) + 4.2063R = 0.2044|R| = 0.4521

2

HAVE PREVENT Data MAX GAP Data

Figure 6-3. LAMARS Combined Phase3 Target Tracking Dataset

6-3

VISTA Combined Datasets

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Sixteen Data Points

Logarithmic Fit

1

2

3

4

5

6

HAVE OLOP Data MAX GAP Data

y = -1.7237 Ln(GC+1 ) + 4.7514R = 0.3267|R| = 0.5716

Figure 6-4. VISTA Combined Phase 2 Sum-of-sines Dataset

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Twelve Data Points

Logarithmic Fit

1

2

3

4

5

6

HAVE OLOP Data MAX GAP Data

y = -0.9114 Ln(GC+1 ) + 3.8760R = 0.1573|R| = 0.3966

Figure 6-5. VISTA Combined Phase 3 Discrete (HUD) Pitch-Tracking Dataset

2

2

6-4

LAMARS and VISTA Combined Dataset Correlation

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Eight Data Points

Sixteen Data Points

Logarithmic Fit1

2

3

4

5

6

HAVE PREVENT LAMARS DataMAX GAP LAMARS DataHAVE OLOP VISTA DataMAX GAPVISTA Data

y = -1.9483 Ln(GC+1 ) + 4.8783R = 0.3137|R| = 0.5601

2

Figure 6-6. LAMARS and VISTA Combined Phase 2 Sum-of-sines Data

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Gap Criterion

PIO

Ten

denc

y R

atin

g

Single Data Point

Two Data Points

Three Data Points

Four Data Points

Five Data Points

Six Data Points

Twelve Data Points

Logarithmic Fit

1

2

3

4

5

6

y = -1.0549 Ln(GC+1 ) + 3.9785R = 0.1793|R| = 0.4234

HAVE PREVENT LAMARS DataMAX GAP LAMARS DataHAVE OLOP VISTA DataMAX GAPVISTA Data

2

Figure 6-7. LAMARS and VISTA Combined Phase 3 Discrete Pitch-Tracking Data

6-5

Combined Dataset Analysis and Observations

Each combined dataset far exceeded a 95% confidence level that correlation

existed. In fact, the lowest correlation confidence level was 99.95% for the LAMARS

Combined Phase 3 Target Tracking. The logarithmic curve fits shown in the previous

figures proved to be the best combination of high correlation factor and endpoint

matching for both high and low Gap Criterion values.

Further inspection of the plots reveals some trends. From the PIO tendency scale

shown in Figure 1-4 of Chapter I, a PIOR of 4, 5 or 6 represents a tendency for PIO while

a PIOR of 1, 2 or3 represents no tendency for PIO, though some undesirable motions are

still possible. From the combined dataset figures in can be seen that PIOR rating is an

inverse function of the Gap Criterion: High PIOR come at low Gap Criterion values and

vice versa. Also, from these figures it can be seen that, in general, the majority of PIO

tendency ratings 4, 5, and 6 occur at Gap Criterion values less than approximately 1.0

while PIO tendency ratings at Gap Criterion values above 1.0 tend to be non-PIO values

of 1, 2 or 3. These two observations were more pronounced in the Phase 2 Sum-of-sines

Tracking Task than in the Phase 3 tasks. This is possibly due to the high gain, high

bandwidth piloting technique used in this task while the other tasks were more focused on

task performances which required both high and low gains and bandwidths.

The Gap Criterion value of 1.0 also has physical significance. Consider a Type I

solution in which G(s) was such that ∆Kp approached zero. Then the Gap Criterion

would simply be the amplitude to cause PIO divided by the maximum amplitude

available and if this Gap Criterion equals 1.0 then the amplitude to cause PIO would be

equal to the maximum amplitude available. Hence, any Gap Criterion value greater than

6-6

1.0 would require more actuator deflection than was available, thereby providing a

natural limit to creating a PIO. In other words, if an amplitude of 35 degrees were

required, with only 30 degrees available, it is readily apparent that a PIO cannot be

achieved.

The relatively high correlation factors, especially for the Combined Phase 2 Sum-

of-sine datasets indicate the potential for Gap Criterion validation and acceptance as a

tool for predicting Category II PIO due to rate-limited actuators.

7-1

VII. Conclusions

In this study, a new criterion for predicting pilot-induced oscillation tendency

rating due to rate-limited actuators was developed and correlated to datasets of these

ratings. This criterion was called the Gap Criterion.

Two historical databases, Projects HAVE PREVENT and HAVE OLOP were

selected to see if the Gap Criterion had merit. Most datasets were assessed with greater

than 95% confidence that a correlation indeed existed. Further, a logarithmic curve fit

was deemed best. Follow on testing in Project MAX GAP gathered more PIO tendency

rating data to augment these earlier findings. These datasets also showed strong evidence

of correlation and again found a logarithmic interpolation of the data worked well.

These datasets were combined for different tasks and sources to determine

whether the Gap Criterion was universal in nature. This seemed to be the case with all

combined datasets indicating greater than 99.95% confidence level that correlation

existed between the PIOR and the Gap Criterion. Logarithmic curve fits again appeared

superior with high relative correlation factors and good endpoint matching.

Further observations were made for the combined datasets. Based on the relative

positions of a majority of the data, it was found that lower Gap Criterion values resulted

in higher PIO tendency ratings and vice versa. Further, proper PIO represented by PIO

tendency rating values of 4, 5 or 6 were clustered at Gap Criterion values of 1.0 or less

while PIO tendency ratings of 1, 2 or 3, representing non-PIO, were more prevalent at

Gap Criterion values greater than 1.0.

The Gap Criterion has merit. It should be used as a tool to predict and reduce

incidents of Category II PIO due to rate-limited actuators during aircraft development.

A-1

Appendix A. Matlab/SimulinkTM Code

This appendix lists the MatlabTM source code used to move the short period poles

of the F-16 to desired locations and then determine the feedback gains for angle-of-attack

and pitch rate necessary to augment the new bare aircraft dynamics and return them to

suitable closed loop dynamics. A SimulinkTM diagram is also shown and was used for

the preceding process. The next code listing computes the Gap Criterion for various bare

aircraft dynamics and rate limit choices.

Bare Aircraft Pole Placement

%Bare Aircraft Dynamics Bare Pole placement %This matlab file will take the bare F-16 dynamics at 15,000 ft Pressure Altitude, %300 KCAS and place the short period poles where I want them based %on what I give for short period damping and natural frequency. clear;clc;format short g; format compact warning off a = [-0.033104 0.14957 -0.3207 -0.56111; -0.015511 -1.2826 1 -0.0024621; 0.008081 -4.0875 -1.7556 0.0012828; 0 0 1 0] b = [-0.5193; -0.05243; -11.085; 0] c = eye(4) d = [0; 0; 0; 0] %Input the Desired Short Period Damping Ratio and Short Period Natural Frequency for test case %Example: MAX GAP LAMARS Case B zetasp = 0.61, omegansp = 2.34 zetasp = input('Short Period Damping Ratio:'); omegansp= input('Short Period Natural Frequency:'); sigma = zetasp*omegansp omegad = omegansp*sqrt(1-zetasp^2) %Compute P vector to "place" the poles for the desired Short Period Damping %Ratio and Short Period Natural Frequency. %Also, Phugoid pole locations are chosen as -0.017+/-0.074j %(Phugoid Damping Ratio = 0.224, Phugoid Natural Frequency = 0.075928 rad/s) P=[-.017+.074*j -.017-.074*j -sigma+omegad*j -sigma-omegad*j] kinner=-1*place(a,b,P) 'ahat is a+b*kinner'

A-2

ahat=a+b*kinner 'eigenvalues of ahat' eigenvalues = eig(ahat) 'natural frequencies and dampings of ahat' [wwn,zz]=damp(ahat) '(Phugoid Damping Ratio = 0.2239, Phugoid Natural Frequency = 0.075928 rad/s)' [num,den]=ss2tf(ahat,b,c,d) BareAcftLongDynamics=tf(num(4,:),den) continuing = input('continue'); %The following are the phugoid and short perios (sp1 & sp2) poles for the DESIRED %(must be changed if desired closed loop characteristics are different) %closed loop system using angle-of-attack (alpha) and pitch rate (q) feedback: phugoid1=-.017+.074*j;phugoid2=-.017-.074*j;sp1=-2.2+2.22*j;sp2=-2.2-2.22*j P2=[phugoid1 phugoid2 sp1 sp2] K=place(ahat,b,P2) %Find approximate values of Kq (call it Kqbase) and %Kalpha (call it Kalphabase) Kqbase=-K(1,3) Kalphabase=-K(1,2) %Find best Kq and Kalpha %This is an iterative, graphical technique to find Kalpha and Kq that take the %bare aircraft dynamics and change them into the desired closed loop dynamics %The error function {E(ii)=abs(wn(3)-3.125)+4.439*abs(z(3)-.7)} is important %in that it relates to the closed loop short period natural frequency desired (3.125) %and the short period damping ratio desired (.7). The constant 4.439 is a weighting funtion %so that frequency and damping ratio are considered equally (3.125/.7 = 4.439) %This is to get a ballpark Kalpha and Kq ii=1 for kq = Kqbase-1:.025:Kqbase+1; for kalpha = Kalphabase-1:.025:Kalphabase+1; Kqcounter(ii)= kq;Kacounter(ii)=kalpha; [A2,B2,C2,D2]=LINMOD('AugmentedDynamics',0); Yc=ss(A2,B2,C2,D2); [Yctfnum,Yctfden]=ss2tf(A2,B2,C2,D2); yctf=tf(Yctfnum,Yctfden); [wn,z]=damp(yctf); wnn(ii)=wn(3);zz(ii)=z(3);iii(ii)=ii; E(ii)=abs(wn(3)-3.125)+4.439*abs(z(3)-.7); ii=ii+1 end end plot(iii,E) grid on grid minor

A-3

xx=input('what index?') 'kq',Kqcounter(xx) 'kalpha',Kacounter(xx) Kqbase=Kqcounter(xx); Kalphabase=Kacounter(xx); %This is to get a more precise Kalpha and Kq ii=1 for kq = Kqbase-.1:.0025:Kqbase+.1; for kalpha = Kalphabase-.1:.0025:Kalphabase+.1; Kqcounter(ii)= kq;Kacounter(ii)=kalpha; [A2,B2,C2,D2]=LINMOD('AugmentedDynamics',0); Yc=ss(A2,B2,C2,D2); [Yctfnum,Yctfden]=ss2tf(A2,B2,C2,D2); yctf=tf(Yctfnum,Yctfden); [wn,z]=damp(yctf); wnn(ii)=wn(3);zz(ii)=z(3);iii(ii)=ii; E(ii)=abs(wn(3)-3.125)+4.439*abs(z(3)-.7); ii=ii+1 end end plot(iii,E) grid on grid minor xx=input('what FINAL index?') %Final results kq=Kqcounter(xx) kalpha=Kacounter(xx) [A2,B2,C2,D2]=LINMOD('AugmentedDynamics',0); Yss=ss(A2,B2,C2,D2); [Ynum,Yden]=ss2tf(A2,B2,C2,D2) Ytf=tf(Ynum,Yden) damp(Ytf)

A-4

Figure A-1. Augmented Dynamics SimulinkTM Model

Gap Criterion Computation

%This Matlab file will be used to take LAMARS Cases B, N, W and Y and compute the Gap Criterion % Maj Joel Witte % 04 Nov 03 %This Matlab file takes the bare aircraft dynamics (Gc) and multiplies by the closed loop modified %Neal-Smith pilot model (Gp). Then the delta gain, gap (or ∆Kp), is calculated and kstar and omega %are determined. %Finally, the Gap Criterion is determined for each max actuator rate. clear;clc;clf;format compact figure(1) % %The following are the parameters for the MAX GAP LAMARS evalutions. % Gc = bare aircraft dynamics %Gp = Neal-smith pilot model %G = bare aircraft dynamics convolved with the Neal-Smith Pilot Model %Choose which case by commenting/uncommenting the parameters %For example, currently Case B will be computed %Case B Gcb=tf([-11.08 -14.37 -0.5277],[1 2.889 5.578 0.2026 0.03157]) Gpb= -.11483*tf([.32483 1],[.00001 1])*tf([5 1],[1 0],'iodelay',.25); G=Gcb*Gpb %Case N % Gcn=tf([ -5.551e-015 -11.09 -14.37 -0.5277],[1 1.912 9.867 0.3439 0.05648]) % Gpn= -.10919*tf([.32699 1],[.00001 1])*tf([5 1],[1 0],'iodelay',.25); % G=Gcn*Gpn %Case W % Gcw=tf([-3.553e-015 -11.09 -14.37 -0.5277],[1 8.512 22.48 0.8031 0.1279]) % Gpw= -.15254*tf([.30311 1],[.00001 1])*tf([5 1],[1 0],'iodelay',.25); % G=Gcw*Gpw

A-5

%Case Y %Gcy=tf([-6.217e-015 -11.08 -14.37 -0.5277],[1 4.21 5.53 0.2071 0.03103]) %Gpy= -.12528*tf([.31318 1],[.00001 1])*tf([5 1],[1 0],'iodelay',.25); %G=Gcy*Gpy W=LOGSPACE(log10(.1),log10(10),1000); ii=0; for kstarr=.1:.001:1; ii=ii+1; kstar(ii)=kstarr; mag_N(ii)=-20*log10(8*kstarr/(pi^2)); ph_N(ii)=(180/pi)*acos(kstarr) - 180; end figure(1) nichols(G,W) grid on hold on plot(ph_N,mag_N,'r') axis([-200 -70 -20 30]) grid on w1=input('what freq (rad/s) range? Low end =') w2=input('high end =') W1=LOGSPACE(log10(w1),log10(w2),1000); [mag_check1,phase_check1] = NICHOLS(G,W1); mag_check=squeeze(mag_check1); phase_check=squeeze(phase_check1); check = polyfit(phase_check,mag_check,9) swoosh = polyfit(ph_N,mag_N,9) jj=1 for phase=-100:-.1:-170; checkmagdB(jj)=20*log10(polyval(check,phase)); checkphase(jj)=phase; jj=jj+1; end jj=1; for phase=-100:-.1:-170; swooshmag(jj)=polyval(swoosh,phase); swooshphase(jj)=phase; jj=jj+1; end dB=swooshmag-checkmagdB; [Gap,index] = min(dB) inc=10^(Gap/20) nichols(G*inc,W)

A-6

hold on axis([-200 -70 -20 30]) omega=input('just touchin omega') [magtouch,phasetouch]=bode(G,omega) ii=1; for kstarr=.1:.001:1; if ph_N(ii)>phasetouch kstar=kstarr; end ii=ii+1; end kstar maxdeg=30 %Compute Gap Criterion for rate limits 15 deg/sec, 30 deg/sec and 60 deg/sec GapCriterion15=inc*(pi*15/(2*omega*kstar*maxdeg)) GapCriterion30=inc*(pi*30/(2*omega*kstar*maxdeg)) GapCriterion60=inc*(pi*60/(2*omega*kstar*maxdeg)) 'done'

B-1

Appendix B. State Space Matrices for Project MAX GAP

This appendix lists the state space matrices for the F-16 dynamics as well as the

bare aircraft dynamics of each case for Projects HAVE PREVENT, HAVE OLOP and

MAX GAP.

F-16 Dynamics a = b = c = d = Modified Matrices The following matrices are the pole placement results (A = a + b*kinner). HAVE PREVENT Case A A = Case B A = Case C A =

-0.033104 0.14957 -0.3207 -0.56111-0.015511 -1.2826 1 -0.0024620.008081 -4.0875 -1.7556 0.001283

0 0 1 0

-0.5193-0.05243-11.085

0

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

0000

-0.033094 0.069282 -0.38266 -0.56125-0.01551 -1.2907 0.99374 -0.0024770.008304 -5.8013 -3.0782 -0.001753

0 0 1 0

-0.033093 0.17889 -0.3122 -0.56093-0.01551 -1.2796 1.0009 -0.0024440.008312 -3.4615 -1.5741 0.005109

0 0 1 0

-0.033114 0.3331 -0.2587 -0.56113-0.015512 -1.2641 1.0063 -0.0024640.007874 -0.16992 -0.43222 0.00081

0 0 1 0

B-2

Case D A = HAVE OLOP Case A A = Case B A = Case C A = Case D A = MAX GAP (LAMARS) Case B A =

-0.033102 0.38576 -0.20764 -0.56119-0.015511 -1.2588 1.0114 -0.002470.008121 0.95415 0.65786 -0.000441

0 0 1 0

-0.035901 -0.21367 -0.22435 -0.56209-0.015793 -1.3193 1.0097 -0.002562-0.051625 -11.841 0.30117 -0.019725

0 0 1 0

-0.033649 0.31011 -0.23615 -0.56149-0.015566 -1.2664 1.0085 -0.0025-0.003556 -0.66055 0.04924 -0.006762

0 0 1 0

-0.033422 0.20283 -0.29951 -0.56139-0.015543 -1.2772 1.0021 -0.002490.001303 -2.9505 -1.3034 -0.004668

0 0 1 0

-0.033278 0.44459 -0.21943 -0.5614-0.015529 -1.2528 1.0102 -0.0024910.004371 2.2099 0.40609 -0.00486

0 0 1 0

-0.033384 0.17915 -0.31228 -0.56137-0.015539 -1.2796 1.0009 -0.0024890.002102 -3.456 -1.5758 -0.004294

0 0 1 0

B-3

Case N A = Case W A = Case Y A = MAX GAP (VISTA) Case B A = Case N A = Case W A =

-0.032474 -0.27041 -0.57362 -0.56097-0.015447 -1.325 0.97446 -0.0024480.021525 -13.052 -7.1545 0.004179

0 0 1 0

-0.034932 -0.080584 -0.26522 -0.56181-0.015696 -1.3058 1.0056 -0.002533-0.030946 -9.0004 -0.57123 -0.013697

0 0 1 0

-0.032397 0.26148 -0.37461 -0.56107-0.01544 -1.2713 0.99456 -0.0024580.023175 -1.6987 -2.9063 0.002109

0 0 1 0

-0.033384 0.18062 -0.31171 -0.56137-0.015539 -1.2795 1.0009 -0.0024890.002103 -3.4248 -1.5638 -0.004301

0 0 1 0

-0.03532 -0.068363 -0.22347 -0.56194-0.015735 -1.3046 1.0098 -0.002546-0.039218 -8.7395 0.31992 -0.016428

0 0 1 0

-0.032282 0.010816 -0.48286 -0.56098-0.015428 -1.2966 0.98363 -0.002449

0.02563 -7.0493 -5.2171 0.0041290 0 1 0

B-4

Case Y A =

-0.032063 0.36636 -0.36584 -0.56099-0.015406 -1.2607 0.99544 -0.002450.030312 0.54004 -2.7192 0.003902

0 0 1 0

C-1

Appendix C. Correlation Computation

Data Reduction and Analysis

The pre-computed Gap Criterion for the selected bare aircraft and rate limit

configuration was plotted with the assigned PIO Tendency Rating. The confidence level

of the data correlation was determined by computing a correlation coefficient, xyr , shown

in equation C-1 (Wheeler and Ganji, 1996:145-155).

11

22 2

1 1

( )( )

( ) ( )

n

i ii

xyn n

i ii i

x x y yr

x x y y

=

= =

− −=

− −

∑

∑ ∑ (C-45)

Where n is the number of data pairs and x and y are the mean values of x and y which

were obtained from:

1 1,

n n

i ii i

x yx y

n n= == =∑ ∑

(C-46)

To determine if there is correlation to a certain degree of confidence, the absolute

value of the correlation coefficient, xyr ,for n data pairs is compared to the minimum

values of the correlation coefficient shown in Table C-1:

(C-2)

(C-1)

C-2

Table C-1. Minimum Values of the Correlation Coefficient for Confidence Level (Wheeler and Ganji, 1996:147)

Confidence Level n 80% 90% 95% 98% 99% 3 0.951 0.988 0.997 1.000 1.000 4 0.800 0.900 0.950 0.980 0.990 5 0.687 0.805 0.878 0.934 0.959 6 0.608 0.729 0.811 0.882 0.917 7 0.551 0.669 0.754 0.833 0.875 8 0.507 0.621 0.707 0.789 0.834 9 0.472 0.582 0.666 0.750 0.798 10 0.443 0.549 0.632 0.715 0.765 11 0.419 0.521 0.602 0.685 0.735 12 0.398 0.497 0.576 0.658 0.708 13 0.380 0.476 0.553 0.634 0.684 14 0.365 0.458 0.532 0.612 0.661 15 0.351 0.441 0.514 0.592 0.641 16 0.338 0.426 0.497 0.574 0.623 17 0.327 0.412 0.482 0.558 0.606 18 0.317 0.400 0.468 0.543 0.590 19 0.308 0.389 0.456 0.529 0.575 20 0.299 0.378 0.444 0.516 0.561 25 0.265 0.337 0.396 0.462 0.505 30 0.241 0.306 0.361 0.423 0.463 35 0.222 0.283 0.334 0.392 0.430 40 0.207 0.264 0.312 0.367 0.403 45 0.195 0.248 0.294 0.346 0.380 50 0.184 0.235 0.279 0.328 0.361 100 0.129 0.166 0.197 0.233 0.257 200 0.091 0.116 0.138 0.163 0.180

D-1

Appendix D. MAX GAP Histograms

The MAX GAP LAMARS and VISTA test histograms are contained in this

appendix. These charts show the raw data collected and indicate which datapoints were

considered outliers and omitted from the reduced dataset (Witte and others, 2003). Pilots

1, 2 and 3 were the same individual in each case.

MAX GAP (LAMARS) Histograms

Handling Qualities Phase 2 Sum-of-sines Task

4

5 5 5

6

5

0

1

2

3

4

5

6

1 1 2 2 3 3Evaluation Pilot

PIO

R


Figure D-1. LAMARS Phase 2 Sum-of-sines Data, Case B, 15 deg/sec

D-2

3

4

5 5

2

5

0

1

2

3

4

5

6


PIO

R



2 2

5

4

5

2

0

1

2

3

4

5

6


PIO

R



Om

itted

Out

lier

Om

itted

Out

lier

D-3

5 5 5 5 5

0

1

2

3

4

5

6

1 2 2 3 3Evaluation Pilot

PIO

R


Figure D-4. LAMARS Phase 2 Sum-of-sines Data, Case N, 15 deg/sec

1

2

5 5

3

5 5

0

1

2

3

4

5

6

1 1 2 2 3 3 3Evaluation Pilot

PIO

R



D-4

1

2

5

3

1

0

1

2

3

4

5

6


PIO

R



4 4 4 4 4 4 4 4

0

1

2

3

4

5

6

1 1 1 2 2 2 3 3Evaluation Pilot

PIO

R


Figure D-7. LAMARS Phase 2 Sum-of-sines Data, Case W, 15 deg/sec

Om

itted

Out

lier

D-5

2

3

4 4 4 4

0

1

2

3

4

5

6


PIO

R



1 1 1 1 1

4

2 2

0

1

2

3

4

5

6


PIO

R



Om

itted

Out

lier

D-6

5 5 5 5 5 5

0

1

2

3

4

5

6


PIO

R


Figure D-10. LAMARS Phase 2 Sum-of-sines Data, Case Y, 15 deg/sec

4 4

5 5

4

5

0

1

2

3

4

5

6


PIO

R



D-7

6

2

5

1

4

1

0

1

2

3

4

5

6


PIO

R



Om

itted

Out

lier

Om

itted

Out

lier

Om

itted

Out

lier

D-8

Handling Qualities Phase 3 Discrete HUD Pitch Tracking Task

5

6

0

1

2

3

4

5

6

2 3Evaluation Pilot

PIO

R


Figure D-13. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case B, 15 deg/sec

4 4

2

0

1

2

3

4

5

6

1 2 3Evaluation Pilot

PIO

R



D-9

3 3 3

0

1

2

3

4

5

6


PIO

R



4

5

0

1

2

3

4

5

6

2 3Evaluation Pilot

PIO

R


Figure D-16. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case N, 15 deg/sec

D-10

4 4 4 4

3

0

1

2

3

4

5

6


PIO

R



2 2

3

0

1

2

3

4

5

6


PIO

R



D-11

4 4 4

2

3

0

1

2

3

4

5

6


PIO

R


Figure D-19. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case W, 15 deg/sec

4

2

4

0

1

2

3

4

5

6


PIO

R



D-12

1

2

3

0

1

2

3

4

5

6


PIO

R



5

3

5 5

0

1

2

3

4

5

6

1 2 3 1Evaluation Pilot

PIO

R


Figure D-22. LAMARS Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 15 deg/sec

D-13

4

3 3

0

1

2

3

4

5

6


PIO

R



4

2

4

0

1

2

3

4

5

6


PIO

R



Om

itted

Out

lier

Om

itted

Out

lier

D-14

Handling Qualities Phase 3 Target Tracking Task

5

2

5

0

1

2

3

4

5

6


PIO

R


Figure D-25. LAMARS Phase 3 Target Tracking Data, Case B, 15 deg/sec

4

2

5

0

1

2

3

4

5

6


PIO

R



D-15

3

1

2

0

1

2

3

4

5

6


PIO

R



5

2

5

0

1

2

3

4

5

6


PIO

R


Figure D-28. LAMARS Phase 3 Target Tracking Data, Case N, 15 deg/sec

D-16

2 2

4

3

0

1

2

3

4

5

6


PIO

R



3

2

0

1

2

3

4

5

6

2 3Evaluation Pilot

PIO

R



D-17

4 4 4

0

1

2

3

4

5

6


PIO

R


Figure D-31. LAMARS Phase 3 Target Tracking Data, Case W, 15 deg/sec

2 2

1

0

1

2

3

4

5

6


PIO

R



D-18

2 2

1

2 2

0

1

2

3

4

5

6


PIO

R



4

5

2

5

0

1

2

3

4

5

6


PIO

R


Figure D-34. LAMARS Phase 3 Target Tracking Data, Case Y, 15 deg/sec

D-19

4

2

5

0

1

2

3

4

5

6


PIO

R



2 2

1

0

1

2

3

4

5

6


PIO

R



D-20

MAX GAP (VISTA) Histograms

Handling Qualities Phase 2 Sum-of-sines Task

5

2

3

5 5 5

4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-37. VISTA Phase 2 Sum-of-sines Data, Case B, 15 deg/sec

3

2

5 5 5

4

5

2

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



Om

itted

Om

itted

Out

lier

D-21

1

5

2

5

1

4

2

3

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



4 4

5 5 5

4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-40. VISTA Phase 2 Sum-of-sines Data, Case N, 15 deg/sec

Om

itted

Out

lier

Om

itted

Out

lier

Om

itted

Out

lier

D-22

1

2

3 3

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



4

3

4

1

5

1 1

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



Om

itted

Out

lier

Om

itted

Out

lier

Om

itted

Out

lier

D-23

4

5 5

4 4 4

5 5

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-43. VISTA Phase 2 Sum-of-sines Data, Case W, 15 deg/sec

4 4

2

5 5

1 1

2

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



D-24

2

1 1 1

2

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



2

5 5 5 5 5

4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-46. VISTA Phase 2 Sum-of-sines Data, Case Y, 15 deg/sec

Om

itted

D-25

1

2

1

5 5

4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-47. VISTA Phase 2 Sum-of-sines Data, Case Y, 30 deg/sec

2

3

1

5

2

3

1

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-48. VISTA Phase 2 Sum-of-sines Test Data, Case Y, 60 deg/sec

Om

itted

Out

lier

D-26

Handling Qualities Phase 3 Discrete HUD Pitch Tracking Task

5

3

4

5

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-49. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case B, 15 deg/sec

2

4

3

2

5

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



D-27

2

3 3

2

4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



4

3

4 4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-52. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case N, 15 deg/sec

D-28

2

3

0

1

2

3

4

5

6

3 3Evaluation Pilot

PIO

Ten

decy

Rat

ing



3 3

4

2 2

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



D-29

4

5

4 4

5

0

1

2

3

4

5

6

1Evaluation Pilot

PIO

Ten

decy

Rat

ing


Figure D-55. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case W, 15 deg/sec

2

5

2 2

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



D-30

2 2

0

1

2

3

4

5

6

3 3Evaluation Pilot

PIO

Ten

decy

Rat

ing



2

4

5

4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing


Figure D-58. VISTA Phase 3 Discrete HUD Pitch Tracking Data, Case Y, 15 deg/sec

Om

itted

D-31

3

2

4

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



2

3 3

2

3

0

1

2

3

4

5

6


PIO

Ten

decy

Rat

ing



BIB-1

Bibliography

Ahlgren, Jan. Report on the Accident Involving the JAS 39-1 Gripen February 2, 1989. T1-1E-89. Government Accident Investigation Board (SHK), Sweden, March 1989. Anderson, Mark R. and Anthony B. Page. Multivariable Analysis of Pilot-in-the-Loop Oscillations. AIAA-95-3203-CP. Reston, VA: American Institute of Aeronautics and Astronautics, Inc., 1995. Brown, Frank and others. Flying Qualities Testing. Air Force Flight Test Center, Edwards AFB CA, 20 February 2002. Department of Defense. Flying Qualities of Piloted Aircraft. MIL-HDBK-1797. Washington: GPO, 19 December 1997. Dornheim, Michael A. “Boeing Corrects Several 777 PIOs,” Aviation Week and Space Technology, 142 (19): 32–33 (1995). Doman, David B. and Lori Ann Foringer. Interactive Flying Qualities Toolbox for Matlab, Version 2.0. Computer Software. USAF Wright Laboratory, Flight Dynamics Directorate, Flight Control Division, Flying Qualities Section, Wright- Patterson AFB OH, August 1996. Duda, Holger. Effects of Rate Limiting Elements in Flight Control Systems – A New PIO- Criterion. AIAA-95-3204-CP. Reston, VA: American Institute of Aeronautics and Astronautics, Inc., 1995. Gilbreath, Greg and others. A Limited Evaluation of a Pilot-Induced Oscillation Prediction Criterion (HAVE OLOP). AFFTC TIM-00-07. Air Force Flight Test Center, Edwards AFB, CA, December 2000. Gilbreath, Gregory P. Prediction of Pilot-Induced Oscillations (PIO) Due to Actuator Rate Limiting Using the Open-Loop Onset Point (OLOP) Criterio. MS Thesis, AFIT/GAE/ENY/01M-02, Graduate School of Engineering and Management, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, March 2001. Hanley, James G. and others. Comparison of Nonlinear Algorithms in the Prevention of Pilot-Induced Oscillations Caused by Actuator Rate Limiting (Project HAVE PREVENT). Air Force Flight Test Center, Edwards AFB CA, December 2002 (ADA410173).

BIB-2

Hanley, James G. A Comparison of Nonlinear Algorithms to Prevent Pilot-Induced Oscillations Caused by Actuator Rate Limiting. MS thesis, AFIT/GAE/ENY/ 03-4. Graduate School of Engineering and Management, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, March 2003 (ADA413992). Hodgkinson, John. Aircraft Handling Qualities. Reston, VA: American Institute of Aeronautics and Astronautics, Incorporated, 1999. Jensen, Clas and others. “JAS 39 Gripen EFCS: How to Deal with Rate Limiting.” Unpublished report. SAAB, Sweden, Undated. Klyde, David H. and others. Unified Pilot-Induced Oscillation Theory, Volume 1: PIO Analysis with Linear and Nonlinear Effective Vehicle Characteristics, Including Rate Limiting. WL-TR-96-3028. Air Force Research Laboratories, Wright- Patterson AFB OH, December 1995. Liebst, Brad S. Class Handout, MECH 629, Aircraft Handling Qualities and Performance. Graduate School of Engineering and Management, Air Force Institute of Technology, Wright-Patterson AFB OH, July 2001. Liebst, Brad S. and others. “Nonlinear Pre-filter to Prevent Pilot-Induced Oscillations Due to Actuator Rate Limiting.” AIAA Journal of Guidance, Control, and Dynamics, Vol. 25, No. 4: 740-747 (July 2002). MatlabTM /SimulinkTM. Versions 6.1 and 6.5. Computer software. The Mathworks, Inc., Natick MA, 2001. McRuer, Duane T. Pilot-Induced Oscillations and Human Dynamic Behavior. NASA Contractor Report 4683. Hawthorne CA: Systems Technology, Inc., 1995 Microsoft Excel. Version 10.3506.3501 SP-1. Computer Software. Microsoft Corporation, Redmond WA, 2001. Mitchell, David G. and Roger H. Hoh. Development of a Unified Method to Predict Tendencies for Pilot-Induced Oscillations. WL-TR-95-3049. Air Force Research Laboratories, Wright- Patterson AFB OH, June 1995. Mitchell, David G. and David H Klyde. A Critical Examination of PIO Prediction Criteria. AIAA-98-4335. Reston, VA: American Institute of Aeronautics and Astronautics, Inc., 1998. Preston, Jeff D. and others. Unified Pilot-Induced Oscillation Theory, Volume 2: Pilot- Induced Oscillation Criteria Applied to Several McDonnell Douglas Aircraft. WL-TR-96-3029. Air Force Research Laboratories, Wright-Patterson AFB OH, December 1995 (ADB212157).

BIB-3

Slotine, Jean-Jacques E. and Weiping Li. Applied Nonlinear Control. Upper Saddle River, New Jersey: Prentice Hall, 1991. Wheeler, Anthony J. and Ahmand R. Ganji. Introduction to Engineering Experimentation. Upper Saddle River, New Jersey: Prentice Hall, 1996. Witte, Joel B. and others. An Investigation Relating Longitudinal Pilot-induced Oscillation Tendency Ratings to Describing Function Predictions for Rate-limited Actuators (Project MAX GAP). Air Force Flight Test Center, Edwards AFB CA, December 2003.

VITA-1

Vita

Major Joel B. Witte was born in Ft Worth, Texas. He graduated from Burleson

High School, Burleson, Texas in 1986. He earned a Bachelor of Science Degree in

Aerospace Engineering from Texas A&M University in 1990.

After graduation, Major Witte entered Undergraduate Pilot Training at Reese

AFB, Texas in January 1992. He received his wings one year later and was assigned to

fly the C-27A at Howard AFB, Panama. His follow-on assignments included flying the

C-141B and C-17A at Charleston AFB, South Carolina and McChord AFB, Washington.

He has accumulated more than 2800 hours flying time.

Major Witte was selected for the joint Air Force Institute of Technology/USAF

Test Pilot School program in February 2001 and began classes at AFIT in September.

After graduation from AFIT/TPS, he will test Special Operations C-130s at Hurlburt

Field, Florida.

Major Witte is a graduate of Test Pilot School, Class 03A – “The Centurions.”

VITA-1

REPORT DOCUMENTATION PAGE Form Approved OMB No. 074-0188

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of the collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to an penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY)

03-2004 2. REPORT TYPE

Master’s Thesis 3. DATES COVERED (From – To)

Apr 2002 – Mar 2004 5a. CONTRACT NUMBER

5b. GRANT NUMBER

4. TITLE AND SUBTITLE 5.

AN INVESTIGATION RELATING LONGITUDINAL PILOT-INDUCED OSCILLATION TENDENCY RATING TO DESCRIBING FUNCTION PREDICTIONS FOR RATE-LIMITED

ACTUATORS

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER 5e. TASK NUMBER

6. AUTHOR(S) Witte, Joel B. Major, USAF

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAMES(S) AND ADDRESS(S) Air Force Institute of Technology Graduate School of Engineering and Management (AFIT/EN) 2950 P Street, Building 640 WPAFB OH 45433-7765

8. PERFORMING ORGANIZATION REPORT NUMBER AFIT/GAE/ENY/04-M16

10. SPONSOR/MONITOR’S ACRONYM(S)

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) USAF TPS 220 Wolfe Ave Edwards AFB CA 93524 DSN: 527-3000 11. SPONSOR/MONITOR’S

REPORT NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

13. SUPPLEMENTARY NOTES 14. ABSTRACT The purpose of this study was to investigate pilot-induced oscillations (PIO) and determine a method by which PIO tendency rating could be predicted. In particular, longitudinal PIO in the presence of rate-limited actuators were singled out for examination. Sinusoidal input/triangular output describing function techniques using Nichols charts were used. A new criterion dubbed Gap Criterion was calculated for PIO sensitivity. This criterion consists of the product of pilot gain necessary to cause PIO and the normalized maximum amplitude of the commanded actuator. These results were paired with simulator and flight test PIO tendency rating data. The PIO rating scale used was the PIO tendency classification of MIL-HDBK-1797. This concept was applied to two historical test databases, HAVE PREVENT and HAVE OLOP. Additional PIO data was gathered in the Large Amplitude Multimode Aerospace Simulator (LAMARS) at the Air Force Research Laboratory (AFRL), Wright-Patterson AFB, Ohio and the USAF NF-16D Variable In-flight Stability Test Aircraft (VISTA) at Edwards AFB, California. Correlation between PIO tendency rating and Gap Criterion was determined for each dataset. Most datasets exceeded a confidence level of 95% that a correlation existed. Follow-on analysis for best curve fit was also accomplished with a logarithmic fit deemed best. Datasets were combined with success to demonstrate the universality of the Gap Criterion for correlating PIO tendency ratings. 15. SUBJECT TERMS handling qualities pilot-induced oscillations pilot-induced oscillation prediction rate limiting flight controls

16. SECURITY CLASSIFICATION OF:

19a. NAME OF RESPONSIBLE PERSON Dr Brad Liebst, AFIT ENY

REPORT U

ABSTRACT U

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VITA-1

Standard Form 298 (Rev: 8-98) Prescribed by ANSI Std Z39-18

AN INVESTIGATION RELATING LONGITUDINAL PILOT-INDUCED ... · 4-1. MAX GAP (LAMARS) Nichols Charts for Cases B, N, W and Y.....4-3 4-2. MAX GAP (LAMARS) Phase 2 Sum-of-sines Task Data.....4-6

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